S⃗eMA JOURNAL
NUMBER 54
April 2011
contents
On the curve straightening flow of inextensible, open, planar curves., by D.B.
Oelz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
On the Modeling of Crowd Dynamics: An Overview and Research Perspectives,
by N. Bellomo, C. Bianca, V. Coscia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson
equations, by Y. Cheng, I.M. Gamba, A. Majorana, C.-W. Shu . . . . . . . . . . . . . 47
On the numerical computation of mountain pass solutions to some perturbed
semi-linear elliptic problem, by L. Montoro . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
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3
S⃗eMA Journal
no 54(2011), 5–24
ON THE CURVE STRAIGHTENING FLOW OF INEXTENSIBLE, OPEN,
PLANAR CURVES.
D.B. OELZ
Faculty of Mathematics, Vienna University
[email protected]
Abstract
We consider the curve straightening flow of inextensible, open, planar curves
generated by the Kirchhoff bending energy. It can be considered as a model
for the motion of elastic, inextensible rods in a high friction regime. We derive
governing equations, namely a semilinear fourth order parabolic equation for the
indicatrix and a second order elliptic equation for the Lagrange multiplier. We
prove existence and regularity of solutions, compute the energy dissipation, prove
its coercivity and conclude convergence to equilibrium, namely to a straight curve,
at an exponential rate.
1
Introduction
We consider the L2 -gradient flow generated by the Kirchhoff bending energy (cp.
[8, 5])
∫
1 1 ′′ 2
E[z] :=
|z | ds
(1)
2 0
on the manifold
A := {z ∈ H 2 ([0, 1], R2 ) : |z ′ | ≡ 1}
Received: January 14, 2010. Accepted: July 14, 2010.
This work has been started during a visit of the author at the Département de mathématiques et
applications (DMA) of the ENS rue d’Ulm Paris within the DEASE project (Marie Curie Early Stage
Training multi Site (mEST) of the EU, MEST-CT-2005-021122). Furthermore it has been supported by
the Vienna Science and Technology Fund (WWTF) through the project ”How do cells move? Mathematical
modelling of cytoskeletal dynamics and cell migration” of C. Schmeiser and V. Small.
The author would like to thank B. Perthame for suggesting the problem and for fruitful discussions and C.
Schmeiser for helpful suggestions.
5
6
D.B. Oelz
of free, open, planar curves of length 1 which are parametrised by their arclength. The
L2 -gradient flow is described by the system (cp. Appendix A)

∂t z + z ′′′′ − (λ1 z ′ )′ = 0 ,


 ′′

 z |s=0,1 = 0 ,
z ′′′ − λ1 z ′ |s=0,1 = 0 ,
(2)

′

|z
|
=
1
,



z(t = 0, .) = zI (.) ,
where λ1 = λ1 (t, s) ∈ R is a Lagrange multiplier function determined by the
constraint on the arclength
|z ′ | ≡ 1
(3)
and zI ∈ A represents the initial datum of the evolution. Here and throughout the
paper ′ denotes derivatives with respect to the arclength.
The system (2) can be considered as a model for the motion of elastic, inextensible
rods in a high friction regime and has appeared in the modelling of Actin-filaments in
biological cells (cp. [19], [18]). Currently the mathematical modelling of biopolymers
and biopolymer networks is a field of high scientific interest and elastic rod models
have recently also been used for the modelling of the DNA (cp [2, 3]). The primary
motivation for this study is to obtain qualitative results on the behaviour of these
models which have the potential to give insight into the behaviour of the respective
biological systems.
Continuing the discussion of the system (2), observe that the centre of mass,
∫1
z ds, is a conserved quantity of solutions to the problem (2). Furthermore the energy
0
of the initial datum is given by
∫
1 1 ′′ 2
EI :=
(4)
|z | ds .
2 0 I
Specialising Theorem 8 and Lemma 4 in [17] implies (see also Appendix A for more
details):
(
)
Theorem 1 Let zI ∈ A, then there exist T > 0 and z ∈ L∞ (0, T ); H 2 (0, 1) ∩
(
)
(
)
C 0,1/8 [0, T ]; C 1 ([0, 1]) ∩ H 1 (0, T ); L2 (0, 1) and λ1 ∈ L2 ((0, T ); M(0, 1)),
satisfying z(t = 0, .) = zI , |z ′ | ≡ 1 and
∫ T∫ 1
[z ′′ · v ′′ + ∂t z · v + λ1 z ′ · v ′ ] ds dt = 0 ,
0
0
for every smooth v : [0, T ] × [0, 1] → R2 .
In the theorem above we use the notation M(0, 1) for the space of Radon measures on
the interval (0, 1). The fact that the existence Theorem 1 is only local in time is due to
the fact that in the more complex setting of [17] additional geometric assumptions on
z are required which cannot be guaranteed for all times. In the present special case (2),
however, these assumptions are redundant and the proof of local existence of the above
theorem can be extended. Here and in the rest of the paper we abbreviate the notation
of function spaces writing the subscripts t for function spaces on {t ∈ [0, ∞)} and the
subscript s for function spaces on {s ∈ [0, 1]}.
7
On the curve straightening flow of inextensible, open, planar curves.
0, 1
0, 1
1
2
Corollary 2 Let zI ∈ A, then there are z ∈ Ht,loc
L2s ∩ Ct 2 L2s ∩ Ct 8 Cs1 ∩ L∞
t Hs
2
and λ1 ∈ Lloc,t Ms such that the pair (z, λ1 ) is a weak solution of (2) satisfying
∫ ∞∫ 1
[z ′′ · v ′′ + ∂t z · v + λ1 z ′ · v ′ ] ds dt
(5)
0
0
for all v ∈ Cc∞ (R+ , Cs∞ ), z(t = 0, .) = zI and
√ the constraint (3) in a pointwise sense.
It holds that ∂t z ∈ L2t L2s with ∥∂t z∥L2t L2s ≤ 2 EI .
Proof. A sketch of the proof, which is presented in detail in [17], can be found in
Appendix A. The crucial difference between the present setting and the more complex
problem in [17] is that the inequality (48) holds without additional terms. Hence the
infinite sum for n = 1, 2, ... is bounded and yields a telescoping series which in the
limit as the step size converges to zero becomes (49), the estimate on ∥∂t z∥L2t L2s . In the present study we show that the problem (2) is equivalent to the system

′
∂t ω + ω ′′′′ − ω ′2 ω ′′ − (ω ′ λ) − ω ′ λ′ = 0 ,


 −λ′′ + ω ′2 λ = ω ′′′ ω ′ + (ω ′′ ω ′ )′ ,
ω ′ , ω ′′ , λs=0,1 = 0 ,



ω(t = 0, .) = ωI (.) .
(6)
where λ = λ1 + ω ′2 and ω = ω(t, s) ∈ R represents the ”indicatrix” of the curve z
(e.g. see [14]) so that
z ′ = (cos(ω), sin(ω)) .
(7)
The reconstruction of the curve z from the indicatrix ω has to be done in such a way,
∫1
that the centre of mass of the initial datum z̄I := 0 zI ds ∈ R2 is conserved,
)
)
∫ 1 ∫ s̃ (
∫ s(
cos(ω(t, s̄))
cos(ω(t, s̄))
z(t, s) = z̄I −
ds̄ ds̃ +
ds̄ .
sin(ω(t, s̄))
sin(ω(t, s̄))
0
0
0
The curvature energy (1), in a abuse of notation, can be written as
∫
1 1 ′ 2
E[ω] :=
(ω ) ds ,
2 0
(8)
such that E[z] = E[ω] if the constraint (3) is satisfied.
In this work we prove
Theorem 3 The solution according to Corollary 2 gives a distributional sense to the
system (6), i.e. for ω being the indicatrix of z and λ = λ1 + ω ′2 it holds that
∫ ∞∫ 1
[
]
−ω∂t ψ − ω ′′′ ψ ′ − ω ′′ (ω ′ )2 ψ − λ′ ω ′ ψ + λ ω ′ ψ ′ ds dt = 0 and (9)
0
∫
0
0
∞
∫
1
[ ′′′ ′
]
−ω ω ϕ + ω ′′ ω ′ ϕ′ + λ′ ϕ′ + λ (ω ′ )2 ϕ ds dt = 0 .
(10)
0
1
1
1
for all ψ ∈ H0,t
L2s ∩ L2t H0,s
and ϕ ∈ L2t H0,s
and λ, ω ′ , ω ′′ 0,1 = 0 a.e. on R+ .
0,1/8 0
Cs
Furthermore It holds that ω ∈ Ct
2
2 2
2 1
with ω ′ ∈ L∞
t Ls ∩ Lt Hs and λ ∈ Lt Hs .
8
D.B. Oelz
The regularity results are better than in the preliminary result Corollary 2, most notably
we get L2 -regularity up to the third spatial derivative of the indicatrix, which partly
corresponds to the fourth spatial derivative of the curve z, and we are able to prove that
the Lagrange multiplier is a function λ ∈ L2t Hs1 . Furthermore we prove:
Theorem 4 Let zI ∈ A, let (z, λ1 ) be a solution of problem (2) according to
Corollary 2 and let (ω, λ) be the corresponding solution to (6) according to Theorem 3,
then the energy dissipation is given by
d
E = −D ,
dt
(11)
in the sense that (11) holds weakly in time. The curvature energy is given by (1) and
(8) respectively and the energy dissipation D too can, in an abuse of notation, be
alternatively written in terms of z or ω,
D = D[z] = D[ω] with
∫ 1
2
D[z] :=
|z ′′′′ − (λ1 z ′ )′ | ds and
0
∫
1
D[ω] :=
(12)
[
]
2
2
(ω ′ ω ′′ + λ′ ) + (ω ′′′ − ω ′ λ) ds .
0
Finally we prove coercivity of (12) with respect to the curvature energy (a Poincaré
type inequality) obtaining the exponential decay of the energy.
Theorem 5 (Poincaré-type) Under the assumptions of Theorem 4, let the energy of
the initial datum be given by (4), then it holds that
E ≤ EI exp(−2π 4 t) ,
where again the curvature energy is alternatively given by (1) and (8).
This result and the regularity statements in Theorem 3 are finally used to prove the
convergence of solutions towards straight lines.
Theorem 6 Under the assumptions of Theorem 4, let the energy of the initial datum
be given by (4), then there is ω∞ ∈ R such that
1/2
∥ω(t, .) − ω∞ ∥L2s ≤ C1 exp(−π 4 t) EI
where C1 :=
√ (1
)
2 π +1 +
3
2π 2
+
3
4
and C2 :=
3/2
+ C2 exp(−3π 4 t) EI
√
9 2+3
2π 2
,
+ 34 .
Curve straightening flows have been investigated intensively since the 1980s. The
paper [11] deals with global in time existence of solutions of the curve straightening
flow of closed curves and with the stability of stationary limit shapes (elasticae). In
[12] this work is generalised to curves in arbitrary Riemannian manifolds and in [10]
and [15] the authors deal with categorising the elasticae. In [14] the development of
self-intersections was investigated and in [16] the focus was on finding a Riemannian
structure that ensures that the curve-straightening flow preserves certain symmetries.
9
In more recent works, [21] and [22], the viewpoint shifted more towards PDEmethods. In [21] a gradient flow acting directly on the indicatrix was investigated and
in [22] the L2 -gradient flow of the square curvature functional was investigated in the
space of smooth closed curves and a fourth order semilinear parabolic equation was
derived as a consequence of the indicatrix representation. The author proves global
in time existence and a convergence result that involves the asymptotic exponential
convergence rate due to the spectrum of the linearised model (cp. [7]). In [9] the
evolution of closed, inextensible curves in R3 under the curve straightening flow with
the same curvature functional as in the present study was considered, but without
making use of the indicatrix representation. In the same way in [20] the motion of
elastic planar closed curves under the additional constraint of area-preservation was
considered. Finally in [13] and in [6] efficient ways to compute the evolution towards
the stationary points called elasticae were investigated.
In contrast to the above mentioned studies, the present one considers open curves as
opposed to closed ones. Although the indicatrix representation was also used in earlier
works ([11, 14, 13, 21, 22]) we consider as new the approach pursued in this paper: to
derive a system of scalar valued equations for the indicatrix and the Lagrange multiplier
and to use it in order to compute, respectively represent the energy dissipation. Doing
so we are able to prove coercivity and, finally, convergence to a straight line at an
exponential rate. These results are derived from the non-linear dynamics and therefore
reveal part of the nature of problem (2), (6) respectively. Most notably the coercivity of
(12) with respect to the square curvature energy is global and leads beyond asymptotic
results which one can get by linearisation at the steady state.
The paper is organised as follows:
In Section 2 we derive formally the system (6) and prove the existence theorem 3.
In Section 3 we show the formal derivation of the energy dissipation and prove the
Theorems 4-6.
Finally, in Section 4, we demonstrate a sequence of numerically computed
snapshots of the evolution under consideration and discuss the observed exponential
convergence rate of the curvature energy.
The paper finishes with an appendix which includes technical computations.
2
Indicatrix formulation
As a consequence of the definition of the indicatrix ω such that (7) holds the curvature
of a curve z ∈ A can be written as
ω ′ = z ′⊥ · z ′′ ,
z∈A.
(13)
Here and in the sequel the superscript ⊥ denotes the rotation of a vector by 90o to the
left, (x, y)⊥ = (−y, x). Frequently we will make use of the relations
z ′′ = ω ′ z ′⊥
and
z ′′⊥ = −ω ′ z ′ ,
(14)
which holds whenever |z ′ | ≡ 1 and which implies the equality of (1) and (8) in this
case. Additionally the initial energy (4) can also be written in terms the indicatrix ωI
∫1
of zI , EI = 12 0 (ωI′ )2 ds .
10
D.B. Oelz
We devote the rest of this section to clarifying the relation between the system
(2) and the equation which we will derive directly from the indicatrix based energy
functional (8).
To this end we extend the definition of the indicatrix to non-admissible curves
z ∈ H 2 ((0, 1); R2 ) with non-singular parametrisation such that |z ′ (s)| > 0 for all
0 ≤ s ≤ 1. Requiring z ′ /|z ′ | = (cos(ω), sin(ω)) we obtain
ω′ =
z ′⊥ · z ′′
.
|z ′ |2
(15)
We use various formulations of the bending energy of such a curve. On the one hand
we rely on the straightforward formulation of the Kirchhoff bending energy (1). On
the other hand we define
)2
)2
∫ (
∫ (
1 1 z ′⊥ · z ′′
1 1
ω′
Ep [z] :=
ds =
ds ,
(16)
2 0
|z ′ |p
2 0
|z ′ |p−2
∫1
which is a generalisation of the square curvature functional E5/2 [z] = 0 (z ′⊥ ·
z ′′ /|z ′ |3 )2 dz that is invariant with respect to reparametrisations of z (see Appendix B).
The functional Ep represents, for every p ∈ R, a notion of bending energy, which
coincides with the Kirchhoff energy (1) and (8) respectively whenever the constraint
(3) is satisfied,
E[z] = E[ω] = Ep [z] if |z ′ | ≡ 1 .
(17)
The energy functional (16) coupled to the constraint (3) defines a gradient flow,
which, in its weak formulation, reads
)
∫ ∞ (∫ 1
∫ 1
∂t z · v ds + δEp [z]v +
λp z ′ · v ′ ds dt = 0 ,
(18)
0
0
0
Cc∞ (R+ , Cs∞ )
for all v ∈
with the Lagrange multiplier function λp = λp (t, s) ∈ R to
enforce the constraint (3) (see Appendix A for the derivation of the variational equation
and Appendix B for the computation of the variation δEp [z]). Its solution coincides
with the solution of the system (2), whereas the strong formulation of (18) is given by

(
)′ ((
) )′
∂t z + ω ′′z ′⊥ + (p − 2)ω ′2 − λp z ′ = 0 ,



 ′ ′′
ω , ω , λp =0,
(19)
s=0,1

′

|z
|
≡
1
,


z(t = 0, .) = zI (.) .
Due to (17) both systems, (2) and (19), describe the same evolution of z and the
existence result Corollary 2 applies to the system (19). However, since the variation
of the functional (16) in non-admissible directions depends on the exponent p, the
Lagrange multipliers λp are different for different p and (19) implies that
λp1 − λp2 = (p1 − p2 )ω ′2
(20)
for different values of p. Furthermore in equation (51) of the Appendix B we show
that non-admissible variations of (1) and (16) coincide when p = 1 which justifies
11
the notation λ1 for the Lagrange-multiplier function in system (2). Summarizing we
obtain
Lemma 7 Let zI ∈ A, let (z, λ1 ) be a weak solution to problem (2) according to
0,1/8 0
2
Cs with ω ′ ∈ L∞
Corollary 2 and let λp = λ1 + (p − 1)ω ′2 , then there is ω ∈ Ct
t Ls
such that ω is an indicatrix of z and (z, ω, λp ) constitute a weak solution of (19)
satisfying (18) for all v ∈ Cc∞ (R+ , Cs∞ ).
Proof. Furthermore the regularity of z ′ according to Corollary 2 allows to identify
0,1/8 0
ω ∈ Ct
Cs up to an additive constant being a multiple of 2π. The rest of the
statement is then an consequence of the discussion above, most notably (20).
The case p = 2 has an especially simple structure, since the forces generated by
the curvature and by the Lagrange multiplier are orthogonal in some sense. This is
reflected by the identities
δE2
· δz = −
δz
∫
1
ω ′′ z ′⊥ · δz ′ ds and
0
∫
δE L
· δz =
δz
1
λz ′ · δz ′ ds ,
(21)
0
∫1
where E L [z] := 0 λ(s)(|z ′ |2 − 1)/2 ds is a potential term such that the effect of the
Lagrange multiplier can be considered as its variation. For this reason we choose the
notation
λ := λ2 .
(22)
It holds that
Lemma 8 (Regularity, a-priori estimates) Let (z, ω, λ) bet a weak solution of (19) in
′ ′′
2 2
the case p = 2 in the sense of Lemma 7, then it holds that ω ′′ ∈ L2t L∞
s , ω ω ∈ Lt Ls ,
2 ∞
′
2 2
′′′
2 2
′
2 2
λ ∈ Lt Ls , ω λ ∈ Lt Ls , ω ∈ Lt Ls and λ ∈ Lt Ls . It also holds that
∫
0
∞
∫
1
0
[
]
2
2
2
(ω ′′′ − λω ′ ) + (λ′ + ω ′ ω ′′ ) ds dt = ∥∂t z∥L2 L2s .
t
Proof. First note that the primitive
∫
0
∞
∫
s
0
2
∫
∂t z ds̃ dt ≤
∞
∞
(∫
0
∫s
0
∂t z ds̃ ∈ L2t L∞
s since
)2
1
|∂t z| ds
∫
∞
dt ≤
0
∫
0
1
|∂t z|2 ds dt < 2 EI
0
by (49). We go back to the integrated version of (19) for p = 2
( ′′ ′⊥ )
 ∫s
− (λz ′ ) = 0 ,
 0 ∂t z ds + ω z
′
ω s=0,1 = 0 ,

z(t = 0, .) = zI (.) .
and obtain
′′
ω = −z
′⊥
∫
·
s
∂t z ds and
0
′
∫
λ=z ·
s
∂t z ds ,
0
(23)
12
D.B. Oelz
2 ∞
which implies ω ′′ ∈ L2t L∞
s and λ ∈ Lt Ls . Moreover we find that
∫
∞
0
∫
1
∫
′ 2
∞
(λω ) ds dt ≤
0
0
∫
2
∥λ∥L∞
s
1
(ω ′ ) ds dt ≤
0
∫ ∞
2
≤ 2 EI
∥λ∥L∞
dt < (2 EI )2 . (24)
s
2
0
′
′′
This computation and an analogous one for ω ω imply
ω ′ λ ∈ L2t L2s
ω ′ ω ′′ ∈ L2t L2s .
and
(25)
We write the weak formulation (18) of problem (19) for p = 2 after two integrations
by parts,
∫
∞
0
∫
1
(
0
)
(
)′
′
∂t z + ω ′′ z ′⊥ − (λz ′ ) · v ds dt+
∫ ∞
[ ′′ ′⊥
]1
+
−ω z · v + λz ′ · v + ω ′ z ′⊥ · v ′ 0 dt = 0 , (26)
0
for all v ∈ Cc∞ (R+ , Cs∞ ).
The uniform estimates we obtained allows to
set v = z ′⊥ ϕ for a testfunction ϕ ∈ D(([0, ∞) × [0, 1]) and to obtain
]
∫∞∫1[
(∂t z)z ′⊥ ϕ + ω ′′′ ϕ − λω ′ ϕ ds dt = 0. Specialising (26) for v =
0
0
z ′ ψ with ψ ∈ D([0, ∞) × [0, 1]) we conclude by an analogous computation
∫∞∫1
[(∂t z)z ′ ψ − ω ′ ω ′′ ψ + λψ] ds dt = 0. Due to (49) and (25) this implies
0
0
z ′⊥ · ∂t z + ω ′′′ − λω ′ = 0 a.e.
and z ′ · ∂t z − ω ′ ω ′′ − λ′ = 0 a.e.,
(27)
hence ω ′′′ ∈ L2t L2s and λ′ ∈ L2t L2s .
Finally (27) implies
∫
∞
0
∫
1
[
(ω ′′′ − λω ′ ) + (λ′ + ω ′ ω ′′ )
2
0
∫
∞
∫
=
0
0
1
[(
z ′⊥ · ∂t z
2
)2
]
ds dt =
+ (z ′ · ∂t z)
2
]
2
ds dt = ∥∂t z∥L2 L2s . (28)
t
Next we will derive the governing equations for the quantities ω(t, s) and λ(t, s).
Taking the derivative of the evolution equation in (19) and explicitly evaluating all the
derivatives we infer
(
∂t z ′ + (ω ′′′′ + 3(p − 2)ω ′2 ω ′′ )z ′⊥ + (ω ′′′ + (p − 2)ω ′3 )(−1)ω ′ z ′ +
)
+ (2p − 5)(ω ′′ ω ′ )′ z ′ + (2p − 5)ω ′′ ω ′ ω ′ z ′⊥ −
− (λ′′p z ′ + (λp ω ′ )′ z ′⊥ + λ′p ω ′ z ′⊥ + λp ω ′ (−1)ω ′ z ′ ) = 0 . (29)
Multiplying by z ′⊥ and z ′ respectively we get
( ′′
)

′′′′
′2 ′′
′ ′
 ∂t ω + ω + (5p − 11) ω ω − ω λp + 2ω(λp = 0 , )
−(ω ′′′ + (p − 2)ω ′3 )ω ′ + (2p − 5)(ω ′′ ω ′ )′ − λ′′p − ω ′2 λp = 0 ,
 ω ′ , ω ′′ , λ p s=0,1 = 0 .
13
(30)
where ∂t ω = z ′⊥ · ∂t z ′ . Observe that λ′′p (0) = λ′′p (1) = 0 as a consequence of the
second equation and the boundary conditions. Let us compare the system for some
“canonical” choices for p. In the case p = 1 we obtain

′′′′
′2 ′′
′′
′ ′
 ∂t ω + ω − 6 ω ω − (ω λ1(+ 2ω λ1 ) =)0 ,
′′′
′3
′
′′ ′ ′
′′
−(ω − ω )ω − 3(ω ω ) − λ1 − ω ′2 λ1 = 0 ,
(31)
 ω ′ , ω ′′ , λ 1 s=0,1 = 0 .
As a consequence of the fact that the variations of E and E1 coincide by (51), the elliptic
equation for λ1 is the one that was found by Koiso ([9]).
In the case p = 2, using the notation (22), we obtain the system (6) and prove the
Theorem 3.
Proof of Theorem 3. We start with a weak solution of (19) satisfying (26) for all
v ∈ Cc∞ (R+ , Cs∞ ) in
the case p = 2 according to Lemma 7. The boundary integrals
imply that λ, ω ′ , ω ′′ 0,1 = 0 a.e. on R+ .
We choose a regularising sequence ηk with supp ηk ⊂ B(0, 1/k) ⊂ R2 and set
v = ηk (t̃ − t, s̃ − s) for (t̃, s̃) ∈ Uk with Uk := [1/k, ∞) × (1/k, 1 − 1/k) obtaining
(
)′
∂t zk + ω ′′ z ′⊥ ∗ ηk − (λz ′ )′ ∗ ηk = 0 ,
)′
(
where zk := z ∗ ηk . We omit the tilde and integrate against − φzk′ + |z1′ |2 ψzk′⊥
k
with φ, ψ ∈ D(R+ × [0, 1]) and k large enough such that Uk covers the support of
both testfunctions,
(
)′
∫∫ [
1
−∂t zk · zk′⊥ ′ 2 ψ −
|zk |
Uk
)′ ]
) (
((
)
′′ ′⊥ ′
′ ′
′⊥ 1
(32)
− ω z
∗ ηk − (λz ) ∗ ηk · zk ′ 2 ψ
ds dt = 0 ,
|zk |
∫∫
((
)
)′
′
−∂t zk · (zk′ φ)′ − ω ′′ z ′⊥ ∗ ηk − (λz ′ )′ ∗ ηk · (zk′ φ) ds dt = 0 .
Uk
Making use of the regularity results in Corollary 2 it holds that zk′ → z ′ and, as a
consequence, also ωk → ω, uniformly on (compact subsets of
) R+ × [0, 1], where ωk
is the indicatrix of zk satisfying zk′ /|zk′ | = cos(ωk ), sin(ωk ) . Using these results we
perform integrations by parts with respect to s and t with the expressions in (32) that
involve ∂t zk and pass to the limit, obtaining
∫∫
∫∫
−∂t zk · (zk′ φ)′ ds dt =
∂t zk′ · zk′ φ ds dt =
Uk
Uk
∫∫
1
=
− |zk′ |2 ∂t φ ds dt → 0 as k → ∞
2
Uk
14
D.B. Oelz
and
∫∫
)′
(
∫∫
1
′⊥ 1
∂t zk′ · zk′⊥ ′ 2 ψ ds dt =
−∂t zk · zk ′ 2 ψ ds dt =
|z |
|zk |
U
Uk
∫ ∫k
∫ ∫k
=
∂t ωk ψ ds dt =
−ωk ∂t ψ ds dt →
Uk
Uk
∞
∫
→
0
∫
1
−ω∂t ψ ds dt as
k→∞,
0
2
where we used that ∂t ωk = ∂t zk′ ·zk′⊥ |z1′ |2 . Making use of the the fact that z ′′ ∈ L∞
t Ls
k
by Corollary 2 and ω ′′′ , λ′ in L2t L2s by Lemma 8 we now pass to the limit k → ∞ in
the remaining expressions of (32) and conclude
∫
∞
∫ 1[
−ω∂t ψ−
(
) (
)]
− ω ′′′ z ′⊥ − λ′ z ′ − ω ′′ ω ′ z ′ − λω ′ z ′⊥ · −z ′ ω ′ ψ + z ′⊥ ψ ′ ds dt = 0 ,
∫ ∞∫ 1
(
) (
)
− ω ′′′ z ′⊥ − λ′ z ′ − ω ′′ ω ′ z ′ − λω ′ z ′⊥ · z ′⊥ ω ′ φ + z ′ φ′ ds dt = 0 .
0
0
0
0
This implies (9), (10) where every term is well defined by Lemma 8 and where we
allow for ψ, ϕ in function spaces in which testfunctions are densily contained.
An interesting structural observation concerning the problem (6) is the following.
Remark 1 Let (ω, λ) be a solution of (6), then, at least on a formal level, the
evolution, represented by ∂t ω, and the Langrange mulitplier satisfy the orthogonality
condition
∫
1
∂t ωλ ds = 0 ,
0
which we obtain integrating the first equation in (6) against λ,
∫
∫
1
∂t ωλ ds =
0
1
(
)
ω ′′′ λ′ + ω ′′ ω ′2 λ + λ′ ω ′ λ − λω ′ λ′ ds dt =
0
∫
∞
∫
=
0
1
(
)
ω ′′′ λ′ + ω ′′ ω ′2 λ ds ,
(33)
0
and integrating the second equation in (6) against ω ′′ ,
∫
1
0
( ′′′ ′
)
ω λ + ω ′2 ω ′′ λ ds =
∫
1
(−ω ′ ω ′′ ω ′′′ + ω ′ ω ′′′ ω ′′ ) ds = 0 .
0
This observation seems to be related to the special structure of forces resulting from
curvature and from pressure/tension (cp (21)) and to the fact that we are dealing with
a constrained gradient flow, i.e. the evolution is tangential to the constrained set.
15
3
Energy dissipation and long time convergence
Testing the first equation in the system (6) formally against ω ′′ and the second one
against λ and taking their difference implies that the energy dissipation is given by
d
d 1
E[ω(t)] =
dt
dt 2
∫
1
ω ′2 ds = −
0
∫
1
[
]
2
2
(ω ′ ω ′′ + λ′ ) + (ω ′′′ − ω ′ λ) ds ≤ 0 .
0
(34)
This can be made rigorous stating that the energy dissipation equality (34) holds
weakly in time (Theorem 4).
Proof of Theorem 4. Here the problem is that we cannot directly set ψ in (9)
equal to ω ′′ , since its time derivative cannot necessarily be interpreted as a function.
Therefore we regularise with respect to t using a sequence of mollifiers (ηk )k=1,2,...
with supp ηk ⊂ [−1/k, 1/k].
∫∞
For t̃ ≥ 1/k we denote by ωk (t̃, s) := (ω ∗t ηk )(t̃, s) = 0 ω(t, s)ηk (t̃ − t) dt the
regularised version of ω and evaluate (9) using ψ(t, s) = ηk (t̃ − t) ωk′′ (t̃, s). The time
integral becomes part of the convolution terms and we find the following expression
which holds pointwise for every t̃,
∫ 1[
(
)
∂t ωk ωk′′ − ωk′′′ ωk′′′ − (ω ′′ (ω ′ )2 ) ∗t ηk ωk′′ −
]
− ((λ′ ω ′ ) ∗t ηk ) ωk′′ + ((λ ω ′ ) ∗t ηk ) ωk′′′ ds = 0 ∀
0
t̃ ≥ 1/k .
Observe that we write the convolution of various expressions and ηk explicitly using
the symbol ∗t .
We omit the tildes and integrate against a test function in time ϑ ∈ D(R+ )
obtaining
∫
∞[
∫
1
∂t ϑ
1/k
0
ωk′2 /2 ds + ϑ
∫ 1[
(
)
−ωk′′′2 − (ω ′′ (ω ′ )2 ) ∗t ηk ωk′′ −
0
] ]
− ((λ′ ω ′ ) ∗t ηk ) ωk′′ + ((λ ω ′ ) ∗t ηk ) ωk′′′ ds dt = 0 , (35)
with k large enough such that(supp ϑ ⊂ [1/k,)∞). Next we use the fact that all the
convolved terms like ωk′ , ωk′′′ , (ω ′′ (ω ′ )2 ) ∗t ηk , etc. converge strongly in L2t,loc L2s to
their original counterparts.
We illustrate this with respect to ω ′′′ . Fix k ∈ N, 1/k < t0 < t1 and 0 ≤ s ≤ 1.
16
D.B. Oelz
Using Fubini’s Theorem and Jensen’s Inequality we find that
∫
t1
2
t0
∫
∫
t0
t1
∫
∫
≤2
=2
t0
∞
ωk′′′ (t, s)2
0
t1
t0
t1
≤
∫
∫
(ωk′′′ − ω ′′′ ) dt̃ =
(∫
∞
ωk′′′ (t̃ − t, s)ηk (t) dt − ω ′′′ (t̃, s)
dt̃ ≤
0
∞
( ′′′
)2
ωk (t̃ − t, s) − ω ′′′ (t̃, s) ηk (t) dt dt̃ ≤
∞
(
0
)
ωk′′′ (t̃ − t, s)2 + ω ′′′ (t̃, s)2 ηk (t) dt dt̃ =
0
∫
)2
∫
t1
t1
ηk (t̃−t) dt̃ dt+2
t0
ω ′′′ (t̃, s)2
∫
t0
∞
0
ηk (t) dt dt̃ ≤ 4∥ω ′′′ ∥2L2 .
t
This result and the strong convergence ωk → ω as k → ∞ for every 0 ≤ s ≤ 1 allows
to apply Lebesgue’s Theorem concluding
∫ 1 ∫ t1
2
′′′
′′′ 2
∥ωk − ω ∥L2 ((t0 ,t1 ))L2s =
(ωk′′′ − ω ′′′ ) dt ds → 0 ,
t
0
t0
hence ωk′′′ → ω ′′′ in L2t,loc L2s . We use this result and the analogous one for the other
convolution terms to pass to the limit as k → ∞ in (35) obtaining
]
∫ ∞[
∫ 1
∫ 1
[ ′′′2
]
∂t ϑ
ω ′2 /2 ds + ϑ
−ω − (ω ′′ ω ′ )2 − λ′ ω ′ ω ′′ + λ ω ′ ω ′′′ ds dt = 0 .
0
0
0
(36)
Setting ϕ = λϑ(t) in (10)
∫ ∞
∫ 1
[ ′′′ ′
]
ϑ(t)
−ω ω λ + ω ′′ ω ′ λ′ + λ′2 + λ2 (ω ′ )2 ds dt = 0 .
0
(37)
0
The difference of (36) and (37) finally yields a weak in time formulation of (34).
As an immediate consequence we obtain
Corollary 9 Let z be a solution of (2) according to Corollary 2 with initial energy EI ,
then it holds that
∫ ∞
∥∂t z∥2L2 L2 =
t
s
D dt = EI .
(38)
0
Furthermore it holds that
∥ω ′′ ∥L2t L∞
, ∥λ∥L2t L∞
s
s
∥ω ′ λ∥L2t L2s , ∥ω ′ ω ′′ ∥L2t L2s
∥λ′ ∥L2t L2s , ∥ω ′′′ ∥L2t L2s
√
EI ,
√
≤
2EI and
√
√
2EI + EI .
≤
≤
Proof . Integrate (34) with respect to time and combine it with (28). Using this
result we go again through the estimates in the proof of Lemma 8.
The energy dissipation equality (34) motivates the definition of [ω] in (12). The
equivalent expression in terms of z,D[z], is then a consequence of
17
Lemma 10 Let (z, λ1 ) be a solution of (2) and let ω be the indicatrix of z and
λ = λ1 + ω ′2 such that (ω, λ) satisfy (6), then it holds that
∫
1
D[ω(t, .)] =
|z ′′′′ (t, s) − (λ1 (t, s)z ′ (t, s))′ | ds a.e. in R+ .
2
0
Proof. We compute iterated derivatives of the constraint (3),
0 = z ′ · z ′′ ,
0 = z ′ · z ′′′ + z ′′ · z ′′ = z ′ · z ′′′ + ω ′2 ,
0 = z ′ · z ′′′′ + z ′′ · z ′′′ + 2ω ′ ω ′′ = z ′ · z ′′′′ + ω ′ z ′⊥ · z ′′′ + 2ω ′ ω ′′ = z ′ · z ′′′′ + 3ω ′ ω ′′ ,
making use of (14) and iterated derivatives of (13),
ω ′′ = z ′⊥ · z ′′′ ,
ω
′′′
=z
′′⊥
·z
′′′
(39)
+z
′⊥
·z
′′′′
′ ′
= −ω z · z
′′′
+z
′⊥
·z
′′′′
′3
=ω +z
′⊥
·z
′′′′
.
We obtain z ′′′′ · z ′ = −3ω ′ ω ′′ and z ′′′′ · z ′⊥ = ω ′′′ − ω ′3 , which imply together with
λ1 = λ − ω ′2 (cp. (20)) that
∫
1
[
]
2
2
(ω ′ ω ′′ + λ′ ) + (ω ′′′ − ω ′ λ) ds =
0
∫
1
=
]
[ ′′′′ ′
(z · z − λ′1 )2 + (z ′′′′ · z ′⊥ − λ1 ω ′ )2 ds =
0
∫
1
=
′′′′ ′ ′
z · z z − λ′1 z ′ + z ′′′′ · z ′⊥ z ′⊥ − λ1 ω ′ z ′⊥ 2 ds =
0
∫
1
=
|z ′′′′ − (λ1 z ′ )′ | ds ,
2
0
where we again used (14).
We finally deal the convergence to equilibrium. First we obtain
Lemma 11 Let ω be a function such that D = D[ω] = 0, then ω ′ ≡ 0, i.e. stationary
points of the energy are straight lines.
Proof. We integrate ω ′ ω ′′ + λ′ = 0 and make use of the boundary conditions on
2
λ to conclude that λ = −ω ′2 /2. Furthermore together with (ω ′′′ − ω ′ λ) = 0 this
implies that ω ′′′ + ω ′3 /2 = 0 and the boundary conditions on ω ′ imply that ω ′ ≡ 0 on
[0, 1].
Furthermore we find the exponential decay of the energy formulated in Theorem 5.
Proof of Theorem 5. The proof uses the fact that the best constant in the Poincarétype inequality
∫ 1
∫ 1
2
v ds ≤ C
v ′2 ds
(40)
0
0
18
D.B. Oelz
Figure 1: Plot of the minimiser of the functional D/E with D/E = 505.917
for v ∈ H01 ((0, 1)) is given by the reciprocal value of the first eigenvalue of the
differential operator v ′′ in that space, C = 1/π 2 . Furthermore we use (39) to obtain
∫
2π E = π
2
1
ω ds ≤
1
=
(z
∫
∫
′2
0
∫
0
′′′
′⊥ 2
ω ′′2 ds =
∫
1
· z ) ds =
0
((z ′′′ − λ1 z ′ ) · z ′⊥ )2 ds ≤
0
1
≤
|z
0
≤
1
2
1
π2
′′′
∫
0
1
(41)
′ 2
− λ1 z | ds ≤
1
′ 2
′′′
(z − λ1 z ′ ) ds = 2 D ,
π
where we used (12). As a consequence, the energy dissipation D is coercive with
respect to the energy and and the following Poincaré-type inequality holds
D/E ≥ 2π 4 .
(42)
Observe that the estimate (42) is not sharp, since in the estimations (41) the inequality
(40) is used twice with the optimal constant C, but with different functions and because
the step from line 2 to line 3 involves adding ((z ′′′ −λ1 z ′ )·z ′ )2 = ((|z ′′ |2 +λ1 ))2 = λ2
to the integrand.
The plot in figure 1 show the numerically found ground state of the exponential
dissipation rate, i.e. the minimiser of the functional D/E. Its rate of convergence is
D/E ≈ 505.917, which still leaves a gap to the constant we found in (42), 2π 4 ≈ 200.
We finally are able to give the proof of Theorem 6.
∫1
Proof of Theorem 6. Let ω(t) := 0 ω(t, s) ds be the mean-value of the indicatrix,
19
then Theorem 5 implies
∥ω(t, .) − ω(t)∥2L2 ≤
2
E[ω(t, .)]
π2
(43)
due to a Poincaré-inequality which is analogous to (40). Let µ(s) := 6s(1 − s) and
let (ηk ) be a regularizing sequence such that supp ηk ⊂ [−1/k, 1/k]. In (9) we set
ψ(t, s) = ηk (t̃ − t)µ(s) with t̃ > 1/k obtaining
d
dt̃
∫
1
ωk µ ds+
0
∫
+
1
[ ′′′ ′
]
−ωk µ − (ω ′′ (ω ′ )2 ) ∗ ηk µ − (λ′ ω ′ ) ∗ ηk µ + (λ ω ′ ) ∗ ηk µ′ ds = 0 .
0
Let 0 < t1 < t2 and integrate with respect to time, omitting the tilde. We pass to the
limit as k → ∞ to get
∫
∫
t2
ω̂(t2 ) − ω̂(t1 ) +
t1
1
[ ′′ ′ 2
]
−ω (ω ) µ − λ′ ω ′ µ + λ ω ′ µ′ ds dt = 0 ,
0
where
∫
ω̂(t) :=
1
ω(t, s)µ(s) ds
0
∫1
is a weighted mean of the function ω(t, .). Observe that 0 −ω ′′′ µ′ ds =
∫ 1 ′′ ′′
ω µ ds = −12(ω ′ (1) − ω ′ (0)) = 0 has canceled due to the boundary values
0
of ω ′ . We infer
∫
|ω̂(t2 ) − ω̂(t1 )| = t2
t1
∫
0
1
[ ′′ ′ 2
]
−ω (ω ) µ − 2λ′ ω ′ µ − λ ω ′′ µ ds dt ≤
]
3 [ ′′ ′
′′
2
∥ω ω ∥L2t L2s ∥ω ′ ∥L2t L2s + 2∥λ′ ∥L2t L2s ∥ω ′ ∥L2t L2s + ∥λ∥L2t L∞
≤
∥ω
∥
,
∞
Lt Ls
s
2
where norms in the dimension t are evaluated on the interval (t1 , t2 ). We now
interprete the data at t =
√ t1 < t2 as initial datum. Doing so Theorem 5 implies
that ∥ω ′ ∥L2 ((t1 ,t2 );L2s ≤ E[ω(t1 )]/(π 2 ) and together with the results of Corollary 9
this implies
[
(
)
]
3 3√
2
3/2
|ω̂(t2 ) − ω̂(t1 )| ≤
2E[ω(t1 , .)] +
+ 1 E[ω(t1 , .)] .
(44)
2 π2
π2
The decay of the energy by Theorem 5 implies that for a given sequence of times (tn )
the sequence ω̂(tn ) is Cauchy and it yields a limit value which we call ω∞ ∈ R. Setting
t1 = t and t2 = tn in (44) and passing to the limit as n → ∞ we obtain
[
(
)
]
2
3 3√
3/2
2E[ω(t, .)] +
+ 1 E[ω(t, .)] .
(45)
|ω̂(t) − ω∞ | ≤
2 π2
π2
20
D.B. Oelz
Furthermore, since
∫1
0
µ(s) ds = 1, it holds that
∫ 1
|ω(t) − ω̂(t)| = ω(t, s)(1 − µ(s)) ds =
0
∫ 1 (
)
∫ s
= ω(t, 0) +
ω ′ (t, s̃) ds̃ (1 − µ(s)) ds =
0
0
∫ 1 (∫ s
)
′
=
ω (t, s̃) ds̃ (1 − µ(s)) ds ≤
∫
0
1
≤
0
(46)
0
|ω ′ (t, s̃)| ds̃ ≤ ∥ω ′ (t, s̃)∥L2s =
√
2E(t) .
Together, (43), (45) and (46) imply
∥ω(t, .) − ω∞ ∥L2s ≤ ∥ω(t, .) − ω(t)∥L2s + |ω(t) − ω̂(t)| + |ω̂(t) − ω∞ | ≤
(
)
(
[
)
]
√
√
1
3 3√
2
3/2
≤ 2
+1
E[ω(t, .)] +
2E[ω(t, .)] +
+ 1 E[ω(t, .)] .
π
2 π2
π2
We finally use Theorem 5 and control the last term by an interpolation between the
smallest and the largest powers of exp(−π 4 t) using x2 ≤ (x + x3 )/2 for x ≥ 0, which
yields the result.
4
Numerics
In the sequence of figures 2 we visualise the numerical solution of the recursive scheme
(47) for τ = 10−6 at various points in time. Information about time, energy and the
dissipation rate are printed in the title of each frame. Observe the decay of energy as
compared to the changes in the dissipation rate, which seems to evolve according to
the following scheme. Initially it drops rapidly, then, when the energy is around E ≈ 7,
it takes its minimum at D/E ≈ 510 and the profile of the curve results to be U-shaped
and similar to the minimiser of the exponential dissipation rate we found numerically
(figure 1). Finally, the exponential rate increases again up to values D/E ≈ 1000. This
can be explained by claiming that at low energy, due to the bounds of Corollary 9, the
linear components of the model (6) dominate. Stripping (6) from its nonlinearities we
obtain ∂t ω̃ + ω̃ ′′′′ = 0 with ω̃ ′ (0) = ω̃ ′ (1) = ω̃ ′′ (0) = ω̃ ′′ (1) = 0. The exponential
dissipation rate of this model in the sense of (42) would be D/E ≈ 1001.13, namely
twice the first eigenvalue of the respective fourth order operator. We remark that a
future study will be devoted to investigating this asymptotic behaviour in more detail.
A
Construction of solutions by the steepest descent flow
The results Theorem 1 and Corollary 2 respectively are obtained from the usual
construction in the theory of gradient flows and steepest descent flows (cp. [4], [1]).
This result indeed is a special case of the existence proof formulated in [17]. It is based
21
t=0.000000 E= 100.00 α=D/E= 677401.0
t=0.000009 E= 63.07 α=D/E= 28698.3
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
0
0.5
−0.5
−0.5
t=0.000099 E= 38.30 α=D/E= 3120.3
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5
0
0.5
−0.5
−0.5
t=0.000699 E= 16.16 α=D/E= 860.1
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
0
0.5
−0.5
−0.5
t=0.005999 E= 0.46 α=D/E= 900.8
0
0.5
0
0.5
t=0.009999 E= 0.01 α=D/E= 996.3
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.5
−0.5
0.5
t=0.001999 E= 7.29 α=D/E= 515.1
0.5
−0.5
−0.5
0
t=0.000299 E= 24.71 α=D/E= 1444.5
0.5
−0.4
0
0.5
−0.5
−0.5
0
Figure 2: Evolution computed numerically.
0.5
22
D.B. Oelz
on defining the recursive scheme
{∫
Zτ0
= zI
Zτn
and
1
∈ argminw∈A
0
[
] }
1 ′′ 2 |w(s) − Zτn−1 |2
ds , (47)
|w | +
2
2τ
where (Zτn )n=0,1,... is a stepwise in time approximation of z with stepsize τ . The
passage to the limit τ → 0 is based on the estimate which we obtain by summing up
for n = 1, 2, ... the fact that
]
∫ 1
∫ 1[
)
1 ( n
1
1
n−1 2
n ′′ 2
Zτ − Zτ
+ ((Zτ ) ) ds ≤
((Zτn−1 )′′ )2 ds .
(48)
2τ
2
2
0
0
Taking the sum and the passage to the limit with respect to τ with interpolations of the
sequence (Zτn )n implies the L2t L2s a priori bound on ∂t z,
∫
0
∞
∫
∫
1
|∂t z| ds dt ≤ 2 EI =
2
0
1
|zI′′ |2 ds .
(49)
0
Furthermore the variational equation of (47) after letting τ → 0 becomes (5) for
variations δz ∈ H 2 ([0, 1]) and the Lagrange multiplier λ1 = λ1 (t, s) to enforce
the constraint |z ′ | ≡ 1. The system (2) is then the strong formulation of (5)
coupled with the constraint. We remark that the ”natural” boundary conditions
z ′′ , z ′′′ − λ1 z ′ |s=0,1 = 0 are equivalent to those in the formulation (2), since taking
the second derivative of the constraint (3) implies |z ′′ |2 + z ′ · z ′′′ = 0 and therefore it
holds that 0 = z ′ · z ′′′ − λ1 |s=0,1 = −|z ′′ |2 − λ1 |s=0,1 = −λ1 |s=0,1 .
B
Variation of the Kirchhoff bending energy
In this appendix we gather some of the variational formulas needed in Section 2. The
energies defined in (1) and (16) coincide, if the constraint (3) is satisfied. This holds
too for the variations of these energies with respect to variations δz ∈ H 2 ([0, 1], R2 )
which are admissible to the constraint (3), i.e. which satisfy z ′ · δz ′ ≡ 0. Variations in
a general direction δz ∈ H 2 ([0, 1], R2 ) might differ, we compute
)(
)
(δω)′
ω′ ′
ω′
′
δEp [z]δz =
− (p − 2) ′ p z · δz
ds =
|z ′ |p−2
|z ′ |p−2
|z |
0
) ( ′⊥
)′
(
)
∫ 1(
ω′
z · δz ′
ω ′2
=
−
(p
−
2)
z ′ · δz ′ ds
′ |2
′ |2(p−2)
′ |2(p−1)
|z
|z
|z
0
∫
1
(
for a curve satisfying |z ′ (s)| > 0 for all 0 ≤ s ≤ 1 making use of the fact that we can
exchange variation and differentiation in the case of δω ′ = (δω)′ and that the variation
of the indicatrix is given by δω[z]δz = z ′⊥ · δz ′ /|z ′ |2 .
Observe that for p = 5/2 the bending energy corresponds to the square curvature
∫1
functional, E5/2 [z] = 0 (z ′⊥ · z ′′ /|z ′ |3 )2 dz and its variation is invariant with respect
to variations that only act as reparametrisations, i.e. δz = z ′ ϕ for test functions
ϕ ∈ D([0, 1]). In fact it holds that δE5/2 [z]z ′ ϕ = 0.
23
Continuing this computation using the assumption that the constraint |z ′ | ≡ 1
holds, the variation of the bending energy reads
∫
1
δEp [z]δz =
ω′
((
z ′⊥ · δz ′
0
[
]1
= ω ′ z ′⊥ · δz ′ 0 +
)′
∫
1
− (p − 2) ω ′ z ′ · δz ′
)
ds =
−ω ′′ z ′⊥ · δz ′ − (p − 2) ω ′2 z ′ · δz ′ ds =
0
∫
)
[
(
]1
= ω ′ z ′⊥ · δz ′ − ω ′′ z ′⊥ + (p − 2) ω ′2 z ′ · δz 0 +
1
( ′′ ′⊥
)′
ω z + (p − 2) ω ′2 z ′ ·δz ds
0
(50)
On the other hand the variation of (1) in a general direction δz but evaluated at a curve
which satisfies the constraint (3) reads
∫
1
′′
∫
′′
z · δz ds =
δE[z]δz =
0
∫
=
1
0
[
1
′′
′ ′
′′
′′
′⊥ ′⊥
′′
|z {z· z} z · δz + z · z z · δz ds =
=0
]1
)′
(
ω ′ z ′⊥ · δz ′′ ds = ω ′ z ′⊥ · δz ′ − ω ′ z ′⊥ · δz +
0
[
(
)
]1
= ω ′ z ′⊥ · δz ′ − ω ′′ z ′⊥ − ω ′2 z ′ · δz 0 +
0
∫
1
(
∫
1
(
ω ′ z ′⊥
)′′
· δz ds =
0
)′
ω ′′ z ′⊥ − ω ′2 z ′ ·δz ds = δE1 [z]δz
0
(51)
and apparently coincides with the variation of E1 .
References
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spaces and in the space of probability measures. Lectures in Mathematics ETH
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Anisotropic elastic model for short dna loops, 2006.
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[6] Gerhard Dziuk, Ernst Kuwert, and Reiner Schätzle. Evolution of elastic curves
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(electronic), 2002.
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D.B. Oelz
[7] Daniel Henry. Geometric theory of semilinear parabolic equations, volume 840
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dünnen elastischen Stabes. J. Reine Angew. Math., 56:285–313, 1859.
[9] Norihito Koiso. On the motion of a curve towards elastica. In Actes de la Table
Ronde de Géométrie Différentielle (Luminy, 1992), volume 1 of Sémin. Congr.,
pages 403–436. Soc. Math. France, Paris, 1996.
[10] Joel Langer and David A. Singer. Knotted elastic curves in R3 . J. London Math.
Soc. (2), 30(3):512–520, 1984.
[11] Joel Langer and David A. Singer. Curve straightening and a minimax argument
for closed elastic curves. Topology, 24(1):75–88, 1985.
[12] Joel Langer and David A. Singer. Curve-straightening in Riemannian manifolds.
Ann. Global Anal. Geom., 5(2):133–150, 1987.
[13] Chun-Chi Lin and Hartmut R. Schwetlick. On the geometric flow of Kirchhoff
elastic rods. SIAM J. Appl. Math., 65(2):720–736 (electronic), 2004/05.
[14] Anders Linnér. Some properties of the curve straightening flow in the plane.
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[15] Anders Linnér. Curve-straightening and the Palais-Smale condition. Trans. Amer.
Math. Soc., 350(9):3743–3765, 1998.
[16] Anders Linnér. Symmetrized curve-straightening. Differential Geometry and its
Applications, 18(2):119 – 146, 2003.
[17] Dietmar Oelz and Christian Schmeiser. Derivation of a model for symmetric
lamellipodia with instantaneous crosslink turnover. preprint, 2009.
[18] Dietmar Oelz and Christian Schmeiser. Simulation of lamellipodia with arbitrary
shape. to be submitted, 2009.
[19] Dietmar Oelz and Christian Schmeiser. Cell mechanics: from single scale-based
models to multiscale modelling, chapter How do cells move? mathematical
modelling of cytoskeleton dynamics and cell migration. Chapman and Hall /
CRC Press, to appear, 2009.
[20] Shinya Okabe. The motion of elastic planar closed curves under the areapreserving condition. Indiana Univ. Math. J., 56(4):1871–1912, 2007.
[21] Yingzhong Wen. L2 flow of curve straightening in the plane. Duke Math. J.,
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1995.
S⃗eMA Journal
no 54(2011), 25–46
ON THE MODELING OF CROWD DYNAMICS: AN OVERVIEW AND
RESEARCH PERSPECTIVES
N. BELLOMO∗ , C. BIANCA∗ , AND V. COSCIA†
∗
†
Department of Mathematics, Politecnico di Torino, Italy,
Department of Mathematics, Università di Ferrara, Italy.
[email protected], [email protected],
[email protected]
Abstract
This paper presents an overview of the mathematical approaches to modeling
crowd dynamics by taking into account the complexity of living systems. The
contents refer to the classical representation scales (microscopic, kinetic, and
macroscopic) and to the mathematical frameworks that can be used for the
modeling approach. The analysis focuses on the kinetic scale and, specifically, on
developments of the mathematical kinetic theory of particles with heterogeneous
behaviors. The existing literature is critically analyzed looking ahead to research
perspectives and, more precisely, to a unified modeling strategy in view also of
swarm dynamics modeling.
A
Introduction
This paper presents an overview, followed by a critical analysis, of the mathematical
literature concerning the modeling of crowd dynamics looking forward to the
challenging objective of depicting, by mathematical equations, the complex dynamics
of swarms. Crowds, swarms, and, to a certain extent, vehicular traffic belong to the
class of complex systems. A complex system can be regarded as a large ensemble of
entities that interact each other by rules, which follow specific strategies, and that have
the ability to communicate and organize their own dynamics according to both their
own individual strategy and their interpretation of that of the others [7].
Received: August 27, 2010. Accepted: November 30, 2010.
NB acknowledges the support by the European Union FP7 Health Research Grant number FP7HEALTH-F4-2008-202047. CB acknowledges the support by the FIRB project RBID08PP3J-Metodi
matematici e relativi strumenti per la modellizzazione e la simulazione della formazione di tumori,
competizione con il sistema immunitario, e conseguenti suggerimenti terapeutici, and by the Compagnia
di SanPaolo, Torino, Italy. VC acknowledges the support by the project ”Environmental effects of vehicular
traffic; Mathematical models and simulations”, University of Ferrara, Italy.
25
26
N. Bellomo, C. Bianca, V. Coscia
A major challenge in the study of complex systems lies in the consideration that the
knowledge of the dynamics and interaction of a few entities is not, in general, sufficient
to describe the collective dynamics of the overall system. A further difficulty resides
on the fact that individual dynamics cannot be observed for every single entity, while
the overall behavior can be described and geometrically interpreted.
The previous considerations lead to conclude that complex systems are difficult to
model or understand at a global level based only on the description of the dynamics of
individual elements. On the other hand, recently, some of the systems cited above have
attracted the attention of scientists involved in the design and construction of models.
The present overview takes advantage of the literature on vehicular traffic, which
has widely developed after the pioneering paper by Prigogine and Hermann [56]
as documented in the review paper [11] and in the collection of papers [9]. A
valuable contribution is offered, from the view point of engineering sciences of traffic
phenomena, by the paper of Daganzo [22], who puts in discussion the continuity
assumptions both of the modeling at the macroscopic scale, based on the paradigms of
continuum mechanics, and of the kinetic theory based on the assumption of continuous
dependence of the distribution function over the state of vehicles at the microscopic
scale. Indeed, he observes that the approach of fluid mechanics refers to thousands of
particles, while only a few vehicles are involved by traffic jams.
Moreover, it is correctly claimed that the modeling approach should take into
account the heterogeneous distribution of the quality of drivers and vehicles. Finally,
it is remarked that models should be characterized by a simple structure, considering
that complex models need a large number of parameters, which may not be identified
in practice. It is plain that these criticisms need to be carefully taken into account
also in the case of crowd modeling. For instance, the comment concerning the
heterogeneous distribution of the quality of interacting entities suggests to use methods
recently developed to model complex systems of the living matter [4, 8]. Moreover,
the modeling needs to take into account additional important characteristics such as,
for instance, a remarkable influence of the geometry and quality of the environment
that contains pedestrians and the transition from normal to panic conditions.
This paper aims at offering to the interested reader not only a review of the state of
the art, but mainly various hints towards research perspectives in a challenging field,
where applied mathematicians generally agree that modeling has not yet reached a
satisfying level and that additional work needs to be developed. This is organized with
three more sections each of them characterized by a well defined objective.
More precisely, Section 2 introduces the concept of scaling and representation
according to the microscopic and macroscopic scales. This section includes
also a review, and critical analysis, of models derived at the microscopic and
macroscopic scales, which refer, respectively, to individually identified pedestrian and
to macroscopic observable quantities such as local density and linear momentum.
Section 3 presents some speculations on the modeling approach by methods
of the generalized kinetic theory. Four mathematical frameworks, selected among
various conceivable ones, are analyzed with the aim of understanding how far they
can act as background paradigms for the derivation of specific models. These
frameworks are derived and critically analyzed to put in evidence positive and negative
properties. Indeed, the modeling strategy starts from the selection of the most
On the Modeling of Crowd Dynamics
27
appropriate mathematical structure. One of them refers to the granular essence of
crowd dynamics, and is based on the discretization of velocity space to overcome the
uncorrect continuity assumptions of the probability distributions used in kinetic theory.
Section 4 further proposes some guidelines toward the modeling by the kinetic
theory of active particles, where the interacting entities are modeled as active particles
with the ability to develop specific strategies that are heterogeneously distributed.
Moreover, an insight to modeling panic conditions is proposed, and finally a brief
overview is given on the modeling of swarms by suitable development of the approach
used to model crowd dynamics.
B
Scaling and Representation Problems
Pedestrian dynamics has not been studied as extensively as vehicular traffic, although
the literature in the field is rapidly developing, as documented in the paper by Helbing
and Molnár [33], Hoogendoorn et al. [40, 41]. The aim of modeling consists in
describing collective and self-organization phenomena from a detailed analysis of the
dynamics at the microscopic scale. As known, the first step of the modeling approach,
for all systems of the real world, consists in the identification of the observation and
modeling scales. Subsequently, for each scale the parameters and the variables to
be used towards modeling need to be identified. Classically, the following types of
descriptions can be considered:
Microscopic description, which refers to entities individually identified. In this case,
their position and velocity identify, as dependent variables of time, the state of the
whole system. Mathematical models are generally stated by systems of ordinary
differential equations.
Macroscopic description, which is used when the state of the system is described
by averaged gross quantities, namely density, linear momentum, and kinetic energy,
regarded as dependent variables of time and space. Mathematical models describe the
evolution of the above variables by systems of partial differential equations.
An alternative to the above two scaling approaches is delivered by the kinetic
theory, which is used when the microscopic state of the interacting entities is still
identified by the position and velocity, but their representation is delivered by a suitable
probability distribution over the microscopic state. Mathematical models describe the
evolution of the above distribution function generally by nonlinear integro-differential
equations. This method is treated in the next sections. This section specifically deals
with the representation and modeling at the two above defined scales. Subsequently
a critical analysis is proposed focusing on the crucial problem of the validation of
models.
Bearing this in mind, let us consider a representation referred to two-space
dimensions, while the generalization to a three-space dimension can be formally
obtained by straightforward calculations. Therefore, let Ω be the closed domain
occupied by the crowd whose boundary is ∂Ω. Let ℓ, nM , and VM , respectively, the
largest dimension (diameter) of the domain Ω, the maximum density corresponding
to the highest admissible packing, and the maximum admissible mean velocity which
may be reached, in average, by pedestrians in free flow conditions. Of course isolated
28
pedestrians can reach a limit velocity Vl larger than VM , and Vl = (1 + µ)VM where
µ > 0 is a constant which depends on the characteristics of the domain as well as on
the type of pedestrian.
T
~ν (P)
P
∂Ω
Ω
Figure 1: – Geometry of the domain Ω occupied by the crowd.
The assessment of the independent variables follows, namely t = tr /TC is the
dimensionless time variable referred to the critical time TC = ℓ/VM necessary to the
fastest pedestrian to move at the maximum mean velocity VM along the diameter ℓ
of Ω; x = xr /ℓ, y = yr /ℓ, v = vr /VM are the dimensionless space and velocity
variables. Accordingly, the representation at the two scales that have been previously
indicated can be defined as follows.
B.1 Modeling at the microscopic scale
Let Ω be the crowd domain occupied by N pedestrians. The microscopic description
of the pedestrian dynamics is represented, for each i-th pedestrian with i ∈ {1, . . . , N },
by the following dimensionless variables:
The position vector xi = xi (t) = (xi (t), yi (t));
The velocity vector vi = vi (t) = (vxi (t), vyi (t)).
Mathematical models are generally stated as a large system of ordinary differential
equations where xi and vi are the dependent variables.
The mathematical structure used to model the dynamics of a system constituted
by a system of N interacting pedestrians is borrowed from the classical Newtonian
mechanics as follows:

dxi


= vi ,


dt
(1)


dvi


= Fi (x1 , . . . , xN , v1 , . . . , vN ) ,
dt
29
which is valid when pedestrians can avoid a discontinuous interaction with the walls.
Here Fi models the interaction of the i-th pedestrian with all the others. In the presence
of time dependent external actions the time can be inserted in the argument of Fi .
The solution of (1) provides the time evolution of position and velocity of
pedestrians. Macroscopic quantities are obtained by suitable average performed either
at fixed time over a suitable space domain or at fixed spatial position over a suitable
time interval. In both cases uncertainties and fluctuations cannot be avoided. Different
modeling approaches correspond to different ways of describing the acceleration term
on the basis of a detailed interpretation of individual behaviors.
An interesting example is the so-called social force model, introduced in [33].
These kinds of models are based on the assumption that interactions among pedestrians
are implemented by using the concept of a social force or social field [44]. Important
examples of social force models can be found in [30, 33, 35], and therein references.
The model proposed by Seyfried et al. [57] deals with pedestrians treated as particles
subject to long-range forces induced by the social behavior of the individuals.
The social force model is able to cover several natural phenomena which occur
during pedestrian movements. For instance:
i) Pedestrians normally choose the fastest route and chase a well defined target T as
visualized in Figure 1;
ii) The concept of desired speed can be introduced to reflect the motivation of the
pedestrians to reach the desired goal with the desired speed;
iii) Pedestrians move with an individual speed, taking into account the situation, sex,
age, handicaps, surroundings, and so on. The speed can be, in this case, assumed to be
Gaussian distributed [36];
iv) Pedestrians keep a certain distance from other pedestrians. The distance depends
on the pedestrian density and walking speed. Suitable repulsive, short-range, potentials
can be introduced to describe these phenomena.
v) The interaction potential can be attractive, for long-range interactions, to model
the aggregation phenomena of pedestrians, who often show a trend to walk in groups.
Once separated (for instance if a pedestrian has to avoid an obstacle), the individual
pedestrians try to reform the group.
The above guidelines define a methodological approach to be followed towards the
derivation of specific models. In general, simplicity of the structure of the model is
needed, namely the design of models should include a small number of parameters to
be identified by experimental data.
Technically different approaches can be developed at the microscopic scale with the
aim of reducing computational complexity. For instance, cellular automata models
have recently been used to simulate pedestrian flows, see among others [15, 29, 49, 50].
The model simulates pedestrians as entities (automata) in cells. The walkway is
modeled as grid cells and a pedestrian is represented as a circle that occupies a cell.
One of the crucial problems of the modeling at the microscopic scale consists
in dealing with a large number of equations and in transferring the microscopic
information to the macroscopic level, namely to physical quantities which can be
possibly observed and measured.
30
B.2 Modeling at the macroscopic scale
Modeling the crowd dynamics at macroscopic scale is far less developed than vehicular
traffic [11]. However, the interest in this specific field is rapidly growing related
to various motivations such as engineering applications, for instance interactions of
crowds and the structures of lively bridges [28, 46, 58, 60, 61]; or the modeling of
panic situations and related safety initiatives. For instance, Coscia and Canavesio [18]
develop calculations related to crowd dynamics on the pilgrims Jamarat bridge (Saudi
Arabia), where every year safety conditions are violated.
Macroscopic modeling has been initiated by Henderson’s pioneering works [36,
37], that proposes a modeling approach by equations related to the kinetic theory of
the homogeneous gas constituted of statistically independent particles in equilibrium
in a two-dimensional space. This approach is also documented in Henderson and
Lyons [38], Henderson and Jenkins [39]. Subsequently, Hughes [42, 43] extended
Henderson’s fluid dynamics approach to allow the introduction of factors related to
human decision and interaction by a model that represents pedestrians as a continuous
density field, where an evolving potential function models the guide of the density field
optimally towards its goal. Additional developments of the pedestrian flow modeling
at the macroscopic scale are due to Hoogendoorn and Bovy [40] and Hoogendoorn et
al. [41], which developed a pedestrian flow model formulated into a user’s dynamic
equilibrium assignments. Additional literature is given in [11].
The macroscopic representation of a system constituted by a large number of
interacting individuals concerns groups of pedestrians rather than individual units. The
macroscopic description may be selected for high density, large scale systems in which
the knowledge of the local behavior of groups is sufficient. In details, the macroscopic
description is defined by the following dimensionless variables:
The local density ρ = ρ(t, x) which is referred to the maximum density nM of
pedestrians;
The mean velocity V = V(t, x) which is referred to VM of pedestrians.
The relationship among the flow rate q, the mean velocity V, and the pedestrian
density ρ is q = ρ V.
The mathematical structure suitable for the derivation of specific models
corresponds to conservation of mass and linear momentum, see Hughes [42] and [43]:

∂t ρ + ∇x · (ρV) = 0 ,

(2)

∂t V + (V · ∇x )V = A[ρ, V] ,
where the dot-product denotes inner-product of vectors and A[ρ, V] is a psychomechanical acceleration acting on pedestrians in the elementary macroscopic volume
of the physical space.
Specific models can be designed referring to the above frameworks. In details:
• First-order models are obtained by mass conservation only linked to a closure of the
equilibrium velocity Ve = Ve (ρ). The guidelines for the derivation of models are
analogous to those followed for vehicular traffic flow modeling [12] with the addition
of the selection of flow direction towards the target of pedestrians, for instance the
31
exit. These models have the advantage of a structural simplicity, however they need
plugging experimental data into the model. Due to their simplicity, first-order models
may be useful for some specific applications as documented in the already cited paper
[18], where simulations are developed in a complex geometry.
The ability of the model to depict real flow conditions encourages further studies in
the field, for instance by inserting in the model an intelligent selection of the optimal
pathways related to local flow conditions as well as the influence of panic conditions.
Specifically, these conditions may attract pedestrians towards overcrowding, which are
avoided in normal flow conditions.
• Second-order models are obtained by both equations (2) with the addition of a
phenomenological relation describing the psycho-mechanic acceleration A[ρ, V]. It
seems quite difficult dealing with higher order models due to problems related to a
correct identification of the energy for a system, where the overall amount of available
energy also depends on the individual ability to develop a strategy.
The acceleration A can be viewed, according to a simple approach, as the
superposition of two contributions due to an adaptation to the mean equilibrium flow
velocity Ve measured in steady uniform flow conditions and to local density gradients.
Both contributions are supposed, in a first approximation, to act along the unit vector
⃗ν = ⃗ν (P) directed from P to the target T, as shown in Figure 1. Namely, individuals
at the point P = (x, y) aim to reach at a destination T = (xT , yT ) along their intended
direction of movement given by the unit vector ⃗ν .
Bearing all above ideas in mind, a brief review of some second-order macroscopic
models is given referring specifically to [10]. In details, the acceleration can be
modeled by the contribution of two terms:
A[ρ, V] = AF [ρ, V] + AP [ρ, V] ,
to be modeled according to the following two assumptions:
2.1. The frictional acceleration AF is proportional to the difference between the actual
velocity V and the mean equilibrium velocity Ve (ρ, ⃗ν ) corresponding to the local
density, that is: AF = cF (ρ) (Ve (ρ, ⃗ν ) − V) ,
2.2. The acceleration AP between pedestrians is determined only by the gradient of
pedestrian density as: AP = −cP (ρ) ∇x ρ .
Several different models of the above terms are proposed in [10], where a
qualitative analysis, focused on the properties of hyperbolic structures, is developed.
Models can be further refined by taking the gradients in the computation of the local
mean velocity, for instance by computing it at a density higher than the real one in the
presence of positive gradients and at a lower one for negative gradients, see [23]. More
in general, A is conditioned by ⃗ν .
It is worth mentioning, in the line of this perspective, that Piccoli and Tosin [54, 55]
have proposed a model of evolution of probability measures occupied by the crowd.
It is an interesting approach which, although developed within the framework of the
macroscopic scale, refers to the dynamics at the lower scale, namely to the strategy
developed by pedestrians.
Their approach can be classified as related to first order models, where the mean
velocity is replaced by a probability measure of the velocity field. The evolution is
32
obtained by adaptation of pedestrians to the density in their visibility zone that is also
related to the target pursued by them, for instance to the exit area [10]. A perspective
idea, however still limited to vehicular traffic, is offered by [12], where the velocity
variable is linked to a probability density which evolves in time.
B.3 Model validation
The validation of models can be based on a suitable comparison between predictions of
the model and empirical data. A preliminary step consists in analyzing the advantages
and disadvantages of the selection, in the modeling approach, of one of the above
scales that may even be different from that used to collect empirical data. Accordingly,
the following ten remarks are selected among several ones to be taken into account as
guidelines of the modeling approach:
i) Crowds are definitely discrete with finite degrees of freedom. However it
is necessary, for practical purposes, that the model allows the computation of
macroscopic quantities;
ii) The flow is not continuous, hence models derived at the macroscopic scale are not
consistent with the classical paradigms of continuum mechanics;
iii) The number of individual entities is not large enough to allow the use of continuous
distribution functions within the framework of the mathematical kinetic theory;
iv) Interactions are not localized, as in the case of classical particles, considering that
pedestrians adapt their dynamics to the flow conditions within their visibility zone;
v) Pedestrians have the ability to modify their dynamics according to specific strategies.
This ability is heterogeneously distributed;
vi) This self-organizing ability and the overall strategy is substantially modified by
environmental conditions, such as the appearance of panic situations;
vii) Mathematical models should include a limited number of parameters related
to well defined physical phenomena, which should be technically identified by
experiments;
viii) Empirical data should not be artificially plugged into mathematical models, which
should reproduce them after an appropriate choice of the parameters;
ix) The output of experiments is very sensitive to environmental conditions;
x) Mathematical models are required to reproduce empirical data in steady uniform
flow conditions, and at least at a qualitative level, emerging phenomena which are
observed in unsteady flow.
Empirical data can be roughly classified into two main categories, namely
quantitative results, for instance focused on steady flow conditions or outlet flows, and
qualitative description of emerging behaviors related to the collective self-organizing
ability of humans and animals. Both types of data can and should be used to validate
models, while their artificial plugging into models should be avoided. Specifically,
the ETH report by Buchmueller and Weidmann [17] provides a rich amount of data
and parameters concerning pedestrian traffic related to walking facilities. The report
is very detailed on specific parameters such as pedestrian dimension and weight,
33
viewed as heterogeneously distributed variables, energy consumption, in different
types of pathways, walking speed, lateral oscillation, flow rates, and other similar
data. The report also analyzes the heterogeneous behavior of individuals related to
age, handicaps, and so on. An important information is delivered by the so-called
velocity and fundamental diagrams, which report the one-directional mean equilibrium
dimensionless velocity Ve = Ve (ρ) and flux qe = Ve (ρ) ρ versus the dimensionless
density ρ. An important test for validation is the analysis of the ability of models to
reproduce these diagrams by transferring the dynamics at the microscopic scale into
the macroscopic behavior in steady uniform flow. This result has been achieved in the
case of vehicular traffic [16, 24], but it still needs to be properly developed in the case
of crowd dynamics.
Experimental activities on crowd emerging phenomena are focused on the selforganizing ability of pedestrians as documented in [34], where a variety of emerging
behaviors are visualized. This type of investigation has some analogy with that related
to swarms that we shall see later. A particularly relevant issue refers to experiments
concerning panic phenomena [31] to capture the substantial difference with respect
to the flow in normal conditions. Crowds exhibit self-organizing abilities in various
circumstances [48], which are not observed in vehicular traffic, while other emerging
phenomena present specific analogies, for instance instabilities that generate phase
transition to jam conditions [52].
A further aspect to be considered is the heterogeneous behavior of drivers and
pedestrians. This aspect is even more critical in the physics of crowds where changes in
the environmental conditions can introduce substantial modifications in the individual
behaviors. Indeed, this is the case of transition from normal to panic conditions, where
individuals loose their trend to the target and are attracted by directions, correctly
or incorrectly, related to escape danger. Modeling has to take into account also the
fact that the human interpretation of danger is not, at least in some cases, correct.
For instance, escaping a danger can be identified by the localization of overcrowded
areas, which constitute additional danger, and a subsequent additional panic. This is
an interesting topic, see [31], which is definitely worth for future research activity to
be properly related to well-defined models.
Finally, let us stress that the modeling of heterogeneous behaviors is lost in the
averaging process induced by the macroscopic approach, while it increases the already
relevant complexity of models at the microscopic scale. This aspect can be consistently
treated by methods of the kinetic theory as we shall see in the following.
C
Models at the Kinetic Scale by Generalized Distribution Function
Methods of the generalized kinetic theory have not yet been systematically applied
to model crowd dynamics although some perspective ideas have been given in [5, 8].
Mathematical models are derived, similarly to the classical kinetic theory, by a balance
of the number of pedestrians, hereinafter also called particles, in the elementary volume
of the phase space. Accordingly, the transport term is equated to the net flow in
such volume as it is computed according to specific models at the microscopic scale.
Different models correspond to different ways of modeling interactions that may be
34
localized or long-range in space. Models may also include the role on the dynamics
of interactions of the dimension of particles. Modeling of individual dynamics should
take into account both interactions among particles and their trend toward a certain
objective, for instance the exit area.
More in details, this section presents four mathematical frameworks, selected
among various conceivable ones, that can act as background paradigms for the
derivation of specific models. Indeed, we claim that the modeling strategy should
include the selection of the most appropriate mathematical structure. These
frameworks are critically analyzed to put in evidence the elements that are useful to the
modeling approach. In general, it is worth mentioning that none of these frameworks
takes into account the heterogeneous behavior of pedestrians. Moreover, transitions
from normal flow to panic conditions involves substantial modifications. Both issues
will be discussed in the next section.
In general, the kinetic theory approach is such that the description of the overall
state of the system is given by the following distribution function normalized with
respect to nM :
f = f (t, x, v) : [0, T ] × Ω × Dv → R+ ,
(3)
where Dv ⊂ R2 is the domain of the velocity variable. If f is locally integrable,
f (t, x, v) dx dv denotes the number of individuals, which, at the time t, are in the
elementary volume of the microscopic states [x, x + dx] × [v, v + dv].
Macroscopic observable quantities are obtained by moments weighted by the
velocity variable. For instance, the dimensionless local density and the local flux
respectively read:
∫
ρ(t, x) =
f (t, x, v) dv ,
(4)
v f (t, x, v) dv .
(5)
Dv
∫
and
q(t, x) =
Dv
C.1
Phenomenological models
Let us first consider the framework characterized by the following structure:
(
)
∂t + v · ∂x + F(t, x) · ∂v f (t, x, v) = Q[f, ρ, ⃗ν ](t, x, v) ,
(6)
where F = F(t, x) models the acceleration applied to pedestrians by the outer
environment, and Q[f, ρ, ⃗ν ] = Q[f, ρ, ⃗ν ](t, x, v) depends on the local distribution
function. Q can be derived on a phenomenological basis, and that can be parameterized
by local macroscopic quantities.
A simple way to model the term Q consists in describing a trend to equilibrium
analogous to the BGK model in kinetic theory:
(
)
Q[f, ρ, ⃗ν ](t, x, v) = cr (ρ) fe (v|ρ, ⃗ν ) − f (t, x, v) ,
(7)
where the rate of convergence cr = cr (ρ) depends on the local density ρ, and fe =
fe (v|ρ, ⃗ν ) denotes the equilibrium distribution function that may be parameterized by
35
the local density ρ and by the direction ⃗ν towards the target T. Models generally
assume F = 0; however, acceleration terms may be imposed by the outer environment,
for instance signalling to accelerate or decelerate that may be useful in the case of rapid
evacuation.
Phenomenological models have the advantage of a structural simplicity. They do
not need modeling interactions at the microscopic scale; a constant trend is assumed
toward an equilibrium velocity prescribed a priori. Such a function, as analyzed
in Section 2, is difficult to measure and should be the output of interactions at the
microscopic scale. Moreover, it should be parameterized with respect to the quality of
the environment.
C.2
Models with long-range interactions
Long-range interactions are important in the modeling of particle interactions
considering that individuals develop their own strategy not only by taking into account
nearby particles, but also long distance ones.
This class of models needs the definition of the following two types of particles
involved in the interaction. Test particles representative of the system, being described
by the distribution function f , and field particles which belong to the visibility zone
∆P of the test particles.
The corresponding structure is as follows:
(∂t + v · ∂x + F(t, x) · ∂v + ∂v F[f ]) f (t, x, v) = 0 ,
(8)
where F[f ] = F[f ](t, x, v) represents the acceleration/deceleration due to the
interactions with the field particles which affect the motion of the test particle. This
term, obtained by summing all actions of the field particles belonging to the visibility
zone ∆P of the test particle P localized in x, see Figure 2, reads:
∫
∫
F[f ](t, x, v) =
∆P
φ(x, x∗ , v, v∗ |ρ, ⃗ν )f (t, x∗ , v∗ ) dx∗ dv∗ .
(9)
Dv
where φ = φ(x, x∗ , v, v∗ |ρ, ⃗ν ) is the positional acceleration/deceleration term
applied by the field particles, localized in x∗ with velocity v∗ , to the test particle,
localized in x with velocity v.
The modeling approach needs the identification of the positional acceleration φ
within a suitable visibility zone. Also in this case, the trend of particles towards the
target has to be taken into account, while local density distribution affects the aforementioned actions.
The advantage of this class of models is that the strategy of individuals is identified
by the term φ, which may be parameterized by ρ and ⃗ν . Therefore the development of
appropriate experiments should be addressed to identify such term. The withdraw is
in the additivity of the action applied by all particles in the visibility zone, while this
is not true in general considering that the strategy developed by the test particle is the
output of the field particles viewed as a whole.
36
T
~ν (P′ )
Obstacle
P′
∆P′
~ν (P)
P
∆P
∂Ω
Ω
Figure 2: – Target T and visibility zones ∆P and ∆P′ of the particles localized in P
and P′ .
C.3
Models with weighted binary interactions
These models are based on the assumption that the intensity of the interaction is
fractionated among the field particles which belong to the visibility zone ∆x of the
test particle localized in x. Accordingly, in order to measure the different intensity
expressed by the field particles localized in x∗ , a weight function w(x, x∗ ) : Ω×∆x →
R+ is introduced such that the intensity interaction expressed by w(x, x∗ ) tails off
when the ’distance’ between the test particle localized in x and the field particle
localized in x∗ increases. Moreover the following identity holds true:
∫
w(x, x∗ ) dx∗ = 1.
∆x
The mathematical framework is derived considering another kind of particle, called
candidate. The interaction rule is as follows: Candidate particles can acquire, in
probability, the state of the test particles after an interaction with field particles, while
test particles loose their state after interactions. The dynamics is described by the
following quantities:
η = η(v∗ , v∗ |ρ(t, x∗ )) is the encounter rate between the candidate particle (or test),
with velocity v∗ (or v) and the field particle with velocity v∗ .
P = P(v∗ → v|v∗ , v∗ , ρ, ⃗ν ) is the probability density that a candidate particle, with
velocity v∗ , ends up into the state v of the test particle after the interaction with the
field particle, with velocity v∗ .
This quantity, defined the table of games, is also assumed to depend on ρ and ⃗ν . In
particular, P satisfies, at fixed ρ ∈]0, 1[, the following condition:
∫
P(v∗ → v|v∗ , v∗ , ρ, ⃗ν ) dv = 1 , ∀ v∗ , v∗ ∈ Dv .
37
(10)
Dv
The corresponding mathematical structure is as follows:
(
)
∂t + v · ∂x + F(t, x) · ∂v f (t, x, v)
∫ ∫ ∫
w(x, x∗ )η(v∗ , v∗ |ρ(t, x∗ ))A(v∗ → v|v∗ , v∗ , ρ, ⃗ν )
=
∆x
Dv
Dv
×f (t, x, v∗ ) f (t, x∗ , v∗ ) dx∗ dv∗ dv∗
∫ ∫
−f (t, x, v)
w(x, x∗ )η(v, v∗ |ρ(t, x∗ ))f (t, x∗ , v∗ ) dx∗ dv∗ . (11)
∆x
Dv
This mathematical structure corresponds to the, so-called, averaged stochastic
games [8]. It is immediate to show that simply by assuming w(x, x∗ ) = δ(x∗ −
x), where δ denotes the Dirac’s delta function, it yields models with long-range
interactions.
A brief account can now be given focusing on the modeling of the term A.
Following [5], particles modify their velocities due to interactions by feeling two
different type of actions:
i) Trend towards a target, for instance the exit zone;
ii) Trend towards the stream of the particles on the visibility zone ∆x of the test
particles;
These trends, namely the modification of velocity induced by them, are related to
α(1 − ρ) in the first case, and to αρ in the second case. More precisely, α(1 − ρ)
and αρ indicate the fraction of particles (pedestrians), which modify their velocity due
to interactions. Therefore, the first trend increases with increasing density, while the
second one with decreasing density. Moreover, α is a parameter modeling the quality
of the environment, in detail α = 0 indicates the worst conditions where the movement
is practically prevented, while α = 1 denotes the best conditions where the movement
is at the easiest level.
The critical analysis of this structure leads to conclusions analogous to that of the
preceding subsection, with the additional advantage of providing a framework that can
be further developed towards modeling nonlinear interactions at least by the weight
function that privileges certain areas with respect to the others.
C.4
Models with discrete set velocity
The contents of this subsection is motivated by the already mentioned criticisms on
the continuity assumption of the distribution function over the microscopic state of
particles. This topic has already been treated in [14, 19, 26] to take into account
the granular nature of vehicular traffic by means of kinetic type models with discrete
velocities. The same approach can be generalized to crowd modeling in a way that
38
the overall state of the system is described by a discrete probability distributions over
groups of particles with velocity within a certain velocity domain. In fact, particles
tend to move in clusters, which can be identified and distinguished from each other by
a discrete set of velocity values. Perspective ideas on this approach are given in [5],
and in the work to appear [6].
It is worth mentioning that this approach is based on different motivations with
respect to the so-called discrete Boltzmann equation that is a crude approximation of
physical reality. Here, the discrete velocity space is used to simulate the noncontinuous
behavior of the distribution function, namely it is used to simulate granular flow.
Borrowing some ideas from [6] the velocity variable is considered in polar coordinates
(v, θ), where v ∈ [0, 1] is the velocity module of v and θ ∈ [0, 2π) is the angle that
a horizontal line forms with the velocity vector v. The following discrete sets are
considered:
Iv = {v1 = 0 , . . . , vi , . . . , vn = Vl } ,
and
Iθ = {θ1 = 0 , . . . , θj , . . . , θm+1 = 2 π} .
Let fij (t, x) = f (t, x, vi , θj ), then the corresponding discrete representation is as
follows:
n ∑
m
∑
f (t, x, v) =
fij (t, x)δ(v − vi ) δ(θ − θj ) .
(12)
i=1 j=1
Macroscopic quantities are obtained by weighted sums, which replace the integrals (4)
and (5).
Application in this present case of the same approach used to derive the
mathematical structure in the continuous case yields:
(
)
∂t + vi · ∂x fij (t, x) = H[f ](t, x)
n
m ∫
∑
∑
ij
=
η[ρ(t, x∗ )]ω(x, x∗ )Phk,pq
(ρ, ⃗ν ) fhp (t, x) fkq (t, x∗ ) dx∗
h,k=1 p,q=1 ∆x
n ∑
m ∫
∑
− fij (t, x)
k=1 q=1
η[ρ(t, x∗ )]ω(x, x∗ )fkq (t, x∗ ) dx∗ ,
(13)
∆x
ij
where Phk,pq
denotes the probability density that a (hk)-candidate particle reaches the
(ij)-state after interaction with a (pq)-field particle. The meaning of the other terms
can be recovered by the framework derived in Subsection 3.3.
The qualitative analysis of this framework leads to the same conclusions addressed
to the preceding framework with the additional advantage of the use of discrete values
of the velocity that appears to be more consistent with the physics of the system
under consideration. However, the smoothness with respect to v is lost therefore the
application of the force F = F(t, x) cannot be considered in the same way of the
previous cases. This framework has the advantage, with respect to that of Subsection
3.3, to use discrete variables that are more suited to capture the granular nature of
the real system. Modeling the application of external actions needs, as we have seen,
the approximation of the derivatives with respect to the discrete velocity variables.
Analogous reasonings are needed when the also the space variable is discrete [13].
D
39
Research perspectives
This final section aims at offering to the interested reader some hints to develop
research activity in the field. The guidelines to pursue such objective can be
summarized, according to the authors’ bias, as follows:
– Crowds need to be regarded as a complex system characterized by some specific
features of living systems. More precisely, the interacting entities have the ability to
develop specific strategies, which depend on the state and localization of the other
entities in their interaction domain. Therefore, interactions are nonlinear and not
additive. –
– This ability is heterogeneously distributed among interacting entities. In other
words, such self-organizing ability is not the same for all individuals, and has to be
regarded as a random variable linked to a probability distribution that might have a
local nature and may be modified by several types of interactions at the microscopic
level. Moreover, modeling individual dynamics does not straightforwardly lead to the
mathematical description of the collective dynamics. –
– Rather than selecting one only modeling scale, the mathematical approach should
also consider the simultaneous interaction of two scales, where the lower scale
modifies the heterogeneous behavior at the higher scale. The lower scale refers to
interactions, which do not follow deterministic rules, but can be classified as stochastic
games. –
Differently from the preceding sections, a mathematical formalization is here
substituted by conceptual reasonings, which wait for a detailed mathematical approach
to be developed within specific research programs. Bearing all above in mind, let
us first consider the problem of modeling the heterogeneous behavior of pedestrians.
This matter can be treated according to the methods of kinetic theory of active
particles theory [8], by introducing an activity variable u in the microscopic state
of pedestrians, called here and after active particles.
The microscopic state is the defined by a tern (x, v, u) with u ∈ [0, 1], where u = 0
corresponds to the worse admissible ability, may be due to handicaps, while u = 1 to
the best one, may be at some athletic level. Therefore, the distribution function is
formally defined as follows:
g = g(t, x, v, u) : [0, T ] × Ω × Dv × [0, 1] → R+ .
(14)
The cases studied in Section 3 correspond to the following approximation:
g = f (t, x, v) × δ(u − u0 ) ,
where δ denotes the Dirac’s function and u0 is the activity expressed by all particles
with the same intensity. In all above cases, macroscopic equations can be computed
simply by adding to Eqs (4) and (5) integration over the activity variable.
The objective of the modeling approach consists in deriving suitable evolution
equations for the distribution function defined in Eq. (14). The mathematical structure
defined in Eq. (13) can be generalized to include the role of this new variable by
40
introducing in the probability density A also the activity variable u. Accordingly, such
density can be formally written as follows:
A = A(v∗ → v, u∗ → u|v∗ , v∗ , u∗ , u∗ , ρ, ⃗ν ) ,
(15)
which, for practical purposes can be factorized as follows:
A = B(v∗ → v|v∗ , v∗ , u∗ , u∗ , ρ, ⃗ν ) × C(u∗ → u|u∗ , u∗ , ρ) .
(16)
More in general, at least two-scales should be considered: the higher concerning
the mechanics of individuals and the lower related to their ability to organize the
dynamics. Interactions at the lower scale modify the activity variable which modifies
the dynamics at the higher scale. The two systems are coupled by the local density
and the activity variable. A general rule is that heterogeneity decreases with increasing
density and disappears in overcrowding conditions, when active particles cannot any
more express their own ability.
A brief account can now be given concerning the modeling of the term A. Particles,
according to the conjecture proposed in [5] modify their velocity at interactions by
feeling two actions: a trend towards the target and an attraction to the velocity of the
active particles in the interaction domain of the candidate particle. These two trends
are modeled by the discrete probability densities in a way analogous to that presented
in Subsection 3.3. In details
ε1 α(1 − ρ),
1 − ε1 α(1 − ρ) ,
(17)
1 − ε2 α ρ ,
(18)
and
ε2 α ρ,
where, in both cases, the former is the fraction of particles that modifies the velocity,
the latter indicates the fraction that keep its velocity. The modification of the velocity
depends also on the velocities of the interacting pair and on the direction of their
velocity with respect to ⃗ν . Moreover, α ∈ [0, 1] is a parameter related to the quality
of the environment, while the parameters ε1 and ε2 model the respective weights of
the two different actions. Analogous reasonings can be developed referring to the
framework with discrete velocities Eq. (13).
Panic conditions induce some modifications in the modeling of interactions. This
topic has been carefully treated for models at the microscopic scale, see [32] and
therein cited references. Some perspective ideas can be given focusing on the dynamics
at the microscopic scale:
i) The value of the activity variable increases, namely active particles are more active
in panic conditions;
ii) In normal conditions the trend towards the target is more significant than the
sensitivity to the flow, namely ε1 is larger than ε2 . The opposite occurs during panic;
iii) In some cases active particles follow the stream even if not useful to escape danger.
The characteristics that have been listed above can be technically taken into account
in the modeling of interactions involving particles, thus generating a model which uses
the same mathematical structure, but with a different choice of the parameters.
41
Let us finally consider the possible development of the mathematical approach
we have presented in this paper to modeling the dynamics of swarms. Definitely,
it is an attractive, and challenging, research perspective that keeps engaging a
growing number of applied mathematicians. The mathematical literature on swarm
modeling shows a rather heterogeneous use of different approaches. Among others,
stochastic differential equations [1], macroscopic equations derived from stochastic
perturbation of individual dynamics [25], modeling swarming patterns [59]. Additional
bibliography can be found in the review paper [11] and in the special issue [9].
A specific characteristic is that the swarm has the ability to express a collective
intelligence related to the environmental conditions, which can evolve by learning
processes. This self-organizing ability amounts also to a typical phenomenon of
swarms concerns the flocking phenomena [20, 21, 47, 53].
The experimental activity on swarms differs from that developed in the case
of crowds. In fact, it is mainly focused on understanding the dynamics of the
interactions corresponding to different animal species [2, 3, 27]. Further, experiments
are addressed to understand emerging behaviors, such as flocking phenomena, break
up and aggregation of swarms, which should be depicted by models. Therefore, the
collection of empirical data is generally focused on qualitative, rather than quantitative,
aspects.
The above reasonings do not claim to be exhaustive, but simply show how
the interest in the field is rapidly growing. This stimulates the interest of applied
mathematicians towards this challenging research field. A few guidelines are given
to support modeling projects:
4.1. Generally, swarms refer to specific animal behaviors, which differ from species to
species and that can be modified by external actions that can induce panic.
4.2. Recent studies [2] conjecture, on the basis of empirical data, that some systems
of animal world develop a common strategy based on interactions depending on
topological rather than metric distances. This definitely is a valuable suggestion to
be used towards modeling.
4.3. The swarm has the ability to express a common strategy, which is a nonlinear
elaboration of all individual contributions, generated by each individual based on the
microscopic state of all the others. In general a swarm has the ability to express a
collective intelligence that is generated by a cooperative strategy [47].
4.4. The above mentioned strategy includes a clustering ability (flocking) that prevents
the fragmentation of the swarm. However, when a fragmentation of the domain
occupied by the swarm occurs, the clustering ability induces an aggregation. The
dynamics of interactions differs in the various zones of the swarm. The activity variable
does not significantly change among individuals. However, it may change due to the
position. For instance, from the border to the center of the swarm. In some cases
position depends on a hierarchy inside the group [20, 51].
Of course, it is necessary considering that interactions among active particles of a
swarm are in three space coordinates, while those of particles of a crowd are defined
over two-space coordinates and the mathematical problems are stated in unbounded
domains with initial conditions with compact support. The solution of problems should
provide the evolution in time of the domain of the initial conditions.
42
Finally, we can claim that the approach of the kinetic theory of active particles is
an appropriate candidate to model the dynamics of crowds and, possibly, swarms. The
most delicate issue is, in our opinion, the modeling of interactions at the microscopic
scale, which are nonlinear, namely nonadditive. Moreover, interactions involve a
limited number of particles almost independently on their localization as far as particles
remain in a certain interaction domain. Various hints have been given in this paper.
Some of them are already involving research teams. Hopefully the reader, interested in
the challenging research field reviewed in this paper, can take advantage of them.
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S⃗eMA Journal
no 54(2011), 47–64
A BRIEF SURVEY OF THE DISCONTINUOUS GALERKIN METHOD FOR
THE BOLTZMANN-POISSON EQUATIONS
YINGDA CHENG∗ , IRENE M. GAMBA∗ , ARMANDO MAJORANA† AND CHI-WANG SHU‡
∗
Department of Mathematics and ICES, University of Texas at Austin,
†
Dipartimento di Matematica e Informatica, Università di Catania,
‡
Division of Applied Mathematics, Brown University
[email protected], [email protected],
[email protected],[email protected]
Abstract
We are interested in the deterministic computation of the transients for
the Boltzmann-Poisson system describing electron transport in semiconductor
devices. The main difficulty of such computation arises from the very high
dimensions of the model, making it necessary to use relatively coarse meshes
and hence requiring the numerical solver to be stable and to have good resolution
under coarse meshes. In this paper we give a brief survey of the discontinuous
Galerkin (DG) method, which is a finite element method using discontinuous
piecewise polynomials as basis functions and numerical fluxes based on upwinding
for stability, for solving the Boltzmann-Poisson system. In many situations, the
deterministic DG solver can produce accurate solutions with equal or less CPU
time than the traditional DSMC (Direct Simulation Monte Carlo) solvers. In order
to make the presentation more concise and to highlight the main ideas of the
algorithm, we use a simplified model to describe the details of the DG method.
Sample simulation results on the full Boltzmann-Poisson system are also given.
A
Introduction
The Boltzmann-Poisson (BP) system, which is a semiclassical description of electron
flow in semiconductors, is an equation in six dimensions (plus time if the device
is not in steady state) for a truly three dimensional device, and four dimensions
for a one-dimensional device. This heavy computational cost explains why the BP
Received: September 15, 2010. Accepted: November 30, 2010.
Support from the Institute of Computational Engineering and Sciences and the University of Texas
Austin is gratefully acknowledged. Research supported for the first author by NSF grant DMS-1016001.
Research supported for the second author by NSF grant DMS-0807712 and DMS-0757450. Research
supported for the third author by PRA 2009 Unict. Research supported for the fourth author by NSF grant
DMS-0809086 and DOE grant DE-FG02-08ER25863.
47
48
Y. Cheng, I.M. Gamba, A. Majorana, C.-W. Shu
system is traditionally simulated by the Direct Simulation Monte Carlo (DSMC)
methods [25]. DSMC methods have the advantage that the increase in computational
cost is not significant with the increase of dimensions. However, the simulation
results are often noisy, and it is difficult to compute transient details (time dependent
states), especially if the probability density function (pdf ) is desired. In recent years,
deterministic solvers to the BP system were considered in the literature, see for
example [23, 28, 3, 2, 4, 5, 6, 24]. These methods provide accurate results which,
in general, agree well with those obtained from DSMC simulations, sometimes at a
comparable or even less computational time. Deterministic solvers have the distinct
advantage in resolving transient details for the pdf. However, the main difficulty of
the deterministic solvers arises from the very high dimensions of the model, making it
necessary to use relatively coarse meshes and hence requiring the numerical solver to
be stable and to have good resolution under coarse meshes. This can be challenging
because under coarse meshes, for a convection dominated problem, the solution may
contain high gradient (relative to the mesh) regions, which may lead to instability if
care is not taken in the design of the algorithm.
One class of very successful numerical solvers for the deterministic solvers of
the BP system is the weighted essentially non-oscillatory (WENO) finite difference
scheme [4, 6]. The advantage of the WENO scheme is that it is relatively simple to
code and very stable even on coarse meshes for solutions containing sharp gradient
regions. However, the WENO finite difference method requires smooth meshes to
achieve high order accuracy, hence it is not very flexible for adaptive meshes.
On the other hand, the Runge-Kutta discontinuous Galerkin (RKDG) method,
which is a class of finite element methods originally devised to solve hyperbolic
conservation laws [17, 16, 15, 14, 18], is a suitable alternative for solving the BP
system. Using a completely discontinuous polynomial space for both the test and
trial functions in the spatial variables and coupled with explicit and nonlinearly stable
high order Runge-Kutta time discretization, the method has the advantage of flexibility
for arbitrarily unstructured meshes, with a compact stencil, and with the ability to
easily accommodate arbitrary hp-adaptivity. For more details about DG scheme for
convection dominated problems, we refer to the review paper [20]. The DG method
was later generalized to the local DG (LDG) method to solve the convection diffusion
equation [19] and elliptic equations [1]. It is L2 stable and locally conservative, which
makes it particularly suitable to treat the Poisson equation.
In recent years, we have initialized a line of research to develop and implement the
RKDG method, coupled with the LDG solution for the Poisson equation, for solving
the full BP system, see [7, 8, 9, 10]. It is demonstrated through extensive numerical
studies that the DG solver produces good resolution on relatively coarse meshes for the
transient and steady state pdf, as well as various orders of moments and I-V curves,
which compare well with DSMC results. Our DG solver has the capability of handling
full energy bands [10] that no other deterministic solver has been able to implement so
far.
In this paper, we give a short survey of the DG solver for the full BP system.
The emphasis is on the algorithm details, explained through a simplified model, and on
sample simulation results to demonstrate the performance of the DG solver. The plan of
the paper is as follows: in Section B, we will introduce the BP system and a simplified
Discontinuous Galerkin method for the Boltzmann-Poisson equations
49
model. In Section C, the DG scheme for this model will be presented. Section D
includes some discussion and extensions of the algorithm. In Section E, we present
some numerical results to show the performance of the scheme. Conclusions and future
work are given in Section F. We collect some technical details of the implementation
of the scheme in the Appendix.
B
The Boltzmann-Poisson system, and a simplified model
The evolution of the electron distribution function f (t, x, k) in semiconductors,
depending on the time t, position x and electron wave vector k, is governed by the
Boltzmann transport equation (BTE) [26]
∂f
1
q
+ ∇k ε · ∇x f − E · ∇k f = Q(f ) ,
∂t
~
~
(1)
where ~ is the reduced Planck constant, and q denotes the positive elementary charge.
The function ε(k) is the energy of the considered crystal conduction band measured
from the band minimum; according to the Kane dispersion relation, ε is the positive
root of
~2 k 2
ε(1 + αε) =
,
(2)
2m∗
where α is the non-parabolicity factor and m∗ the effective electron mass. The electric
field E is related to the doping density ND and the electron density n, which equals
the zero-order moment of the electron distribution function f , by the Poisson equation
∇x [εr (x) ∇x V ] =
q
[n(t, x) − ND (x)] ,
ε0
E = −∇x V ,
(3)
where ε0 is the dielectric constant of the vacuum, εr (x) labels the relative dielectric
function depending on the semiconductor and V is the electrostatic potential. For low
electron densities, the collision operator Q(f ) is
∫
Q(f )(t, x, k) =
[S(k′ , k)f (t, x, k′ ) − S(k, k′ )f (t, x, k)] dk′ ,
(4)
R3
where S(k′ , k) is the kernel depending on the scattering mechanisms between
electrons and phonons in the semiconductor.
In order to more clearly describe the details of the DG method, as well as to
highlight the essential algorithm ingredients, we introduce a simplified model transport
equation, which has the same characteristics as the full BP system. Thus, we consider
the system
∫
∂u
∂
∂
+
[a(v) u] +
[b(v) η(t, x) u] =
K(v, v ′ ) u(t, x, v ′ ) dv ′ − ν(v) u, (5)
∂t
∂x
∂v
R
[
] ∫
∂ϕ
∂ϕ
∂
σ(x)
=
u(t, x, v ′ ) dv ′ − ND (x), η(t, x) = − . (6)
∂x
∂x
∂x
R
Now, the unknown distribution function is u, which depends on time t, space
coordinate x ∈ [0, 1] and the variable v ∈ R (velocity or energy). The functions a,
50
b, σ, ND and the kernel K are given, and K ≥ 0. The collision frequency ν is defined
by the equation
∫
K(v ′ , v) dv ′
ν(v) =
(7)
R
which guarantees mass conservation.
C
The DG solver for the simplified model
We now discuss the DG solver for the model equations (5)-(6). The first step is to
reduce the domain of the variable v to a finite size. This can be justified because we
expect a vanishing behavior of the unknown u for large values of v. For rigorous
justification, one could refer to the discussion in [12]. If I is the finite interval
used for the computation in the variable v, then we will replace K(v, v ′ ) with
K(v, v ′ ) χI (v) χI (v ′ ), where χI is the characteristic function on I. The new collision
frequency is redefined according Eq. (7). Therefore, this adjustment of the domain will
not affect the mass conservation of the system.
For simplicity of discussion, we will use a simple rectangular grid to introduce the
DG scheme, although the algorithm could be easily adjusted to accommodate general
unstructured grids. For the domain [0, 1] × I, we let
[
] [
]
Ωik = xi− 12 , xi+ 12 × vk− 12 , vk+ 21
where,
xi± 12 = xi ±
∆xi
2
vk± 12 = vk ±
∆vk
2
(i = 1, 2, 3, ...Nx ) and (k = 1, 2, 3, ...Nv ) .
We denote by Nx and Nv the number of intervals in the x and v direction, respectively.
The approximation space is defined as
Vhℓ = {vh : (vh )|Ωik ∈ P ℓ (Ωik )},
(8)
where P ℓ (Ωik ) is the set of all polynomials of degree at most ℓ on Ωik . Notice that the
polynomial degree ℓ can actually change from cell to cell (p-adaptivity), although in
this paper it is kept as a constant for simplicity. The DG formulation for the simplified
Boltzmann equation (5) would be: to find uh ∈ Vhℓ , such that
∫
∫
∫
(uh )t vh dx dv −
a(v)uh (vh )x dx dv −
b(v)η(t, x)uh (vh )v dx dv
Ωik
Ωik
Ωik
]
∫ [∫
+Fx+ − Fx− + Fv+ − Fv− =
K(v, v ′ ) uh dv ′ − ν(v) uh vh dx dv (9)
Ωik
I
for any test function vh ∈ Vhℓ . In (9),
∫ v 1
k+
2
Fx+ =
a(v) ǔh vh− (xi+ 12 , v)dv,
vk− 1
2
Fx− =
∫
∫
vk+ 1
2
vk− 1
a(v) ǔh vh+ (xi− 12 , v)dv,
2
xi+ 1
2
Fv+ =
Fv− =
∫
51
xi− 1
b(vk+ 12 )η(t, x) ũh vh− (x, vk+ 12 )dx,
2
xi+ 1
2
xi− 1
b(vk− 21 )η(t, x) ũh vh− (x, vk− 12 )dx,
2
where the upwind numerical fluxes ǔh , ũh are chosen according to the following rules,
• if a(v) ≥ 0 on the interval [vk− 12 , vk+ 21 ], ǔh = u−
h ; if a(v) < 0 on the interval
+
[vk− 21 , vk+ 12 ], ǔh = uh . Since a(v) is a given function that does not depend on
time, we will always be able to choose the grid such that a(v) holds constant
signs in each cell [vk− 12 , vk+ 12 ].
∫x 1
+
• If x i+12 b(vk+ 12 )η(t, x)dx > 0, ũh = u−
h ; otherwise, ũh = uh . Since the
i−
2
function η depends on time, we can not choose a grid such that b(vk+ 12 )η(t, x)
holds constant sign on each interval [xi− 12 , xi+ 12 ]. Here we relax the condition
to look at the cell averages for the coefficient for easy implementation.
As for the Poisson equation (6), in the simple one-dimensional setting, one can
use an exact Poisson solver or alternatively use a DG scheme designed for elliptic
equations. Below we will describe the local DG methods [19] for (6) with Dirichlet
boundary conditions. First, the Poisson equation is rewritten into the following form,


 q = ∂ϕ
∂x
(10)
∂


(σ(x)q) = R(t, x)
∂x
∫
′
where R(t, x) = I uh (t, x, v )dv ′ −ND (x) is a known function that can
[ be computed
]
at each time step once uh is solved from (9). The grid we use is Ii = xi− 12 , xi+ 12 ,
with i = 1, . . . , Nx , which is consistent with the mesh for the Boltzmann equation.
The approximation space is
Whℓ = {vh : (vh )|Ii ∈ P ℓ (Ii )},
with P ℓ (Ii ) denoting the set of all polynomials of degree at most ℓ on Ii . The LDG
scheme for (10) is given by: to find qh , ϕh ∈ Vhℓ , such that
∫
∫
qh vh dx +
ϕh (vh )x dx − ϕ̂h vh− (xi+ 12 ) + ϕ̂h vh+ (xi− 21 ) = 0,
Ii
Ii
∫
∫
\
\ +
1 ) − σ(x)q p (x
1) =
−
σ(x)qh (ph )x dx + σ(x)q h p−
(x
R(t, x)ph dx, (11)
i+ 2
i− 2
h h
h
Ii
Ii
hold true for any vh , ph ∈ Whℓ . In the above formulation, the flux is chosen as follows,
+
−
+
\
ϕ̂h = ϕ−
h , σ(x)q h = (σ(x)qh ) − [ϕh ], where [ϕh ] = ϕh − ϕh . At x = L we need
52
−
\
to flip the flux to ϕ̂h = ϕ+
h , σ(x)q h = (σ(x)qh ) − [ϕh ] to adapt to the Dirichlet
boundary conditions. Solving (11), we can obtain the numerical approximation of
the electric potential ϕh and electric field ηh = −qh on each cell Ii . The so-called
minimum dissipation LDG method [13] can also be used here.
To summarize, the DG-LDG algorithm advances from tn to tn+1 in the following
steps:
∫
Step 1 Compute I uh (t, x, v ′ ) dv ′ and R(t, x).
Step 2 Solve the electric field ηh (t, x) from (11).
Step 3 Solve (9) and get a method of line ODE for uh .
Step 4 Evolve this ODE by proper time stepping from tn to tn+1 , if partial time step
is necessary, then repeat Step 1 to 3 as needed.
The algorithm described above carries the essential ideas of those in [9] for the BP
system. We include some of the technical details of the scheme in the Appendix for
the reference of the readers.
D
Some considerations
In this section, we will review some aspects of the DG-BP solver developed in the
literature and give an overview of some ongoing and future research directions.
D.1 Mass conservation and positivity of the numerical solution
It is well known that the transport Boltzmann equation associated with the BP system
conserves mass, and the initial value problem propagates positivity. In particular, it is
essential that the numerical schemes preserve these physical properties of the system.
In general, high order than one DG solvers as the one introduced in the previous
section will not enjoy positivity. However it was shown in [12] that the semi-discrete
DG scheme is positivity-preserving and stable for piecewise constants by arguments
following an adaptation of the Crandall-Tartar lemma [22] to low order DG schemes,
which states that any mass preserving, contracting linear first order operator is stable
and monotone preserving. In addition, in [12] the authors also proposed a fully discrete
positivity-preserving DG scheme for Vlasov-Boltzmann transport equation. They used
a maximum-principle-satisfying limiter for conservation laws [32] and achieved a high
order accurate DG solver that ensures the positivity of the numerical solution. A
comparison study of the standard RKDG scheme against the positivity-preserving DG
scheme has also been performed for the linear Boltzmann equation in [12].
Similarly for the DG-BP solver, clearly the following two properties will also hold,
where the second one is a simple proof of positivity for the fully discrete scheme on
piecewise constant basis functions.
Property 1 (Semi-discrete mass conservation) Under zero or periodic boundary
conditions in the x space, we have that
∫ ∫
d 1
uh dvdx = 0.
dt 0 I
53
Proof. Plug in (9) with vh = 1, and sum over all i, k, the conservation will follow.
Property 2 (Positivity for the first order scheme) Define A = max a(v), B =
max b(v)η(t, x) and V̄ = max ν(v). For first order DG scheme with piecewise
constant approximation and forward Euler time discretization, if the CFL condition
λ1 A + λ2 B + △tV̄ ≤ 1,
and λ2 = min△t
, then the DG scheme is monotone
is satisfied, where λ1 =
k △vk
and will preserve the positivity of the numerical solution.
△t
mini △xi
Proof. Plug in (9) with vh = 1, we have
∫
∫
(uh )t dΩ + Fx+ − Fx− + Fv+ − Fv− =
Ωik
Ωik
∫
where
[∫
]
K(v, v ′ ) uh dv ′ − ν(v) uh dΩ, (12)
I
vk+ 1
2
Fx+ =
a(v) ǔh dv,
vk− 1
Fx−
∫
Fv+
Fv−
∫
2
vk+ 1
2
=
2
xi+ 1
2
=
∫
a(v) ǔh dv,
vk− 1
xi− 1
b(vk+ 21 )η(t, x) ũh dx,
2
xi+ 1
2
=
xi− 1
b(vk− 21 )η(t, x) ũh dx.
2
We assume uh = unik on Ωik at tn , then
un+1
ik
△t
−
=
{F + − Fx− + Fv+ − Fv− +
△xi △vk x
∫ v 1
k+
2
−△xi
ν(v) dv unik }.
∫
∫
unik
Ωik
K(v, v ′ ) uh dv ′ dΩ
I
vk− 1
2
As suggested by the definition for the numerical fluxes, ∫if a(v) ≥ 0 on the
vk+ 1
+
n
2
interval [vk− 12 , vk+ 12 ], ǔh = u−
h , which implies Fx = ( vk− 1 a(v)dv) uik and
2
∫v 1
∫v 1
Fx− = ( v k+12 a(v)dv) uni−1,k . Otherwise, Fx+ = ( v k+12 a(v)dv) uni+1,k and Fx− =
k−
k−
2
2
∫v 1
( v k+12 a(v)dv) unik .
k−
2
∫x 1
= u−
Fv+
=
Similarly, if x i+12 b(vk+ 12 )η(t, x)dx > 0, ũh
h,
i−
2
∫ xi+ 1
∫
x
1
( x 12 b(vk+ 12 )η(t, x) dx) unik and Fv− = ( x i+12 b(vk+ 12 )η(t, x) dx) uni,k−1 ; otheri−
i−
2
2
∫x 1
∫x 1
wise, Fv+ = ( x i+12 b(vk+ 12 )η(t, x) dx) uni,k+1 and Fv− = ( x i+12 b(vk+ 12 )η(t, x) dx) unik .
i−
2
i−
2
By taking derivative of all its variables, it is not difficult to verify that this is a
monotone scheme if
λ1 A + λ2 B + △tV̄ ≤ 1.
Hence it will preserve the positivity of the numerical solution.
54
D.2 Incorporation of full energy band models
The BP system introduced in Section B uses analytical band structures, which means
the energy band function ε(k) has been given explicitly. The analytical band makes
use of the explicit dependence of the carrier energy on the quasimomentum, which
significantly simplifies all expressions as well as implementation of these techniques.
However, the physical details of the band structure are partly or totally ignored, which
is unphysical when hot carriers in high-field phenomena are considered.
Full band models [21], on the other hand, can guarantee accurate physical pictures
of the energy-band function. They are widely used in DSMC simulators, but only
recently the Boltzmann transport equation was considered [31, 29], where approximate
solutions were found by means of spherical harmonics expansion of the distribution
function f . Since only a few terms of the expansion are usually employed, high
order accuracy is not always achieved [27]. Recently in [10], the authors developed
a DG code, which is the first deterministic code that can compute the full band model
directly. The energy band is treated as a numerical input that can be obtained either by
experimental data or the empirical pseudopotential method. The Dirac delta functions
in the scattering kernels can be computed directly, based on the weak formulations of
the PDE. The results in [10] for 1D devices have demonstrated the importance of using
full band model when accurate description of hydromoments under large applied bias
is desired. In a forthcoming manuscript [11], we will present a full implementation
of the scheme with numerical bands as well as a thorough study of stability and error
analysis.
In future work, we will extend the solver to include multi-carrier transport in
devices such as P-N junctions. When modeling the P-N junctions, the carrier flows
of both electrons and holes should be considered. The pdf of electrons and holes will
satisfy the following BTEs coupled with Poisson equation for the field,
∂fi
1
q
+ ∇k εi · ∇x fi ∓ E · ∇k fi = Qi (f ) + Ri (fe , fh ), i = e, h1, h2, h3,
∂t
~
~
q
[ne (t, x) − nh (t, x) − ND (x) + NA (x)] , E = −∇x V .
[εr (x) ∇x V ] =
ε0
In the above equation, the subscript e denotes electrons and h1, h2, h3 denote holes.
One electron conduction band and three hole valence bands (heavy, light and split-off
bands) need to be considered in order to have an accurate physical description. The
Qi terms are the collision terms which have been introduced in Section 2 for electrons.
For holes, those terms should include inter-band scattering as well. Ri (fe , fh ) is the
recombination term. In this model, we need to solve for each carrier a Boltzmann
transport equation and they are all coupled together through the Poisson equation for
the electric field. It will be of particular interest to explore adaptive DG methods for
solving this type of systems in order to reduce computational cost.
E
Numerical results for semiconductor devices
In this section, we will demonstrate the performance of DG schemes through the
calculation for a 2D double gate MOSFET device. The schematic plot of the double
55
gate MOSFET device is given in Figure 1. The top and bottom shadowed region
denotes the oxide-silicon region, whereas the rest is the silicon region.
y
50nm
50nm
top gate
2nm
1111111111111111
0000000000000000
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
x
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
00000000000000000000000000000000000000000000
11111111111111111111111111111111111111111111
0000000000000000
1111111111111111
00000000000000000000000000000000000000000000
11111111111111111111111111111111111111111111
drain
source
24nm
11111111111111111111111111111111111111111111
00000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000
11111111111111111111111111111111111111111111
bottom gate
150nm
Figure 1: Schematic representation of a 2D double gate MOSFET device
Since the problem is symmetric about the x-axis, we will only need to compute
for y > 0. At the source and drain contacts, we implement the same boundary
condition as proposed in [6] to realize neutral charges. At the top and bottom of the
computational domain (the silicon region), we impose the classical elastic specular
boundary reflection. The electric potential Ψ = 0 at source, Ψ = 1 at drain and Ψ =
0.5 at gate. For the rest of boundaries, we impose homogeneous Neumann boundary
condition. The relative dielectric constant in the oxide-silicon region is εr = 3.9,
in the silicon region is εr = 11.7. The doping profile has been specified as follows:
ND (x, y) = 5×1017 cm−3 if x < 50nm or x > 100nm, ND (x, y) = 2×1015 cm−3 in
the channel 50nm ≤ x ≤ 100nm. All numerical results are obtained with a piecewise
linear approximation space and second order TVD Runge-Kutta time stepping. We use
a very coarse mesh, 24 × 12 grid in space, 24 points in w, 8 points in µ and 6 points
in φ in our calculation. In Figures 2 and 3, we show the results of the macroscopic
quantities for the top part of the device when it is already at equilibrium.
56
8e+17
7e+17
6e+17
5e+17
4e+17
3e+17
2e+17
1e+17
0
0.00 0.03
0.06 0.09
x
0.12 0.15
0.024
0.018
0.012
y
0.006
0.3
0.25
0.2
0.15
0.00 0.03
0.06 0.09
x
0.12 0.15
1.4e+07
1.3e+07
1.2e+07
1.1e+07
1e+07
9e+06
8e+06
0.00 0.03
0.06 0.09
x
0.12 0.15
0.024
0.018
0.012
y
0.006
0.024
0.018
0.012
y
0.006
1e+06
500000
0
-500000
-1e+06
0.00 0.03
0.06 0.09
x
0.12 0.15
0.024
0.018
0.012
y
0.006
Figure 2: Macroscopic quantities of double gate MOSFET device at t = 0.8ps. Top
left: density in cm−3 ; top right: energy in eV ; bottom left: x-component of velocity in
cm/s; bottom right: y-component of velocity in cm/s. Solution reached steady state.
0
-20
-40
-60
-80
-100
-120
-140
0.00 0.03
0.06 0.09
x
0.12 0.15
50
0
-50
-100
-150
-200
-250
-300
-350
0.00 0.03
0.06 0.09
x
0.12 0.15
1
0.8
0.6
0.4
0.2
0
0.00 0.03
0.06 0.09
x
0.12 0.15
57
0.024
0.018
0.012
y
0.006
0.024
0.018
0.012
y
0.006
0.024
0.018
0.012
y
0.006
Figure 3: Macroscopic quantities of double gate MOSFET device at t = 0.8ps. Top
left: x-component of electric field in kV /cm; top right: y-component of electric field
in kV /cm; bottom: electric potential in V . Solution has reached steady state.
58
F Concluding remarks and future work
In this paper, we present a brief survey of the current state-of-the-art of DG solvers for
BP systems in semiconductor device simulations. We demonstrate the main ideas of the
algorithm through a simplified model. We include some discussions of the properties
and extensions of the schemes and show numerical results for the full BP system.
Deterministic solvers have recently gained growing attention in the field of
semiconductor device modeling because of the guaranteed accuracy and noise-free
simulation results they provide. However, the relative cost of this type of methods is
still large especially when the dimension of the device is high. From the stand point of
algorithm design and development, it will be interesting to explore ways to utilize fully
the freedom of the DG framework such as hp-adaptivity. The DG schemes also provide
an excellent platform for potential areas such as the hybridization of different level of
models. For practical purposes, we will develop kinetic models and solvers to simulate
nano-scale devices such as bipolar transistors, heterostructure bipolar transistors and
solar cells in the future. Parallel implementation will also be an important component
of our future research.
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discontinuous Galerkin methods for elliptic problems, SIAM Journal on
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[2] M.J. Caceres, J.A. Carrillo, I.M. Gamba, A. Majorana and C.-W. Shu,
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60
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61
G Appendix
In this appendix, we will include some of the implementation details of the proposed
DG algorithm for the full BP system.
For a silicon device, the collision operator (4) takes into account acoustic
deformation potential and optical intervalley scattering [30, 33]. For low electron
densities, it reads
∫
Q(f )(t, x, k) =
[S(k′ , k)f (t, x, k′ ) − S(k, k′ )f (t, x, k)] dk′
(13)
R3
with the scattering kernel
S(k, k′ )
(nq + 1) K δ(ε(k′ ) − ε(k) + ~ωp )
+ nq K δ(ε(k′ ) − ε(k) − ~ωp ) + K0 δ(ε(k′ ) − ε(k))
=
(14)
and K and K0 being constant for silicon. The symbol δ indicates the usual Dirac
distribution and ωp is the constant phonon frequency. Moreover,
[
(
)
]−1
~ωp
−1
nq = exp
kB TL
is the occupation number of phonons, kB the Boltzmann constant and TL the constant
lattice temperature. In Table 1, we list the physical constants for a typical silicon
device.
m∗ = 0.32 me
K=
K0 =
(Dt K)2
8π 2 ϱ ωp
kB TL
Ξ2
4π 2 ~ v02 ϱ d
εr = 11.7
TL = 300 K
~ωp = 0.063 eV
Dt K = 11.4 eV Å−1
ϱ = 2330 kg m−3
Ξd = 9 eV
v0 = 9040 m s−1
α = 0.5 eV
Table 1. Values of the physical parameters
For the numerical treatment of the Boltzmann-Poisson system (1), (3), it is convenient
to introduce suitable dimensionless quantities and variables. Typical values for length,
time and voltage are ℓ∗ = 10−6 m, t∗ = 10−12 s and V∗ = 1 Volt, respectively. Thus,
we define the dimensionless variables
(x, y, z) =
x
,
ℓ∗
t=
t
,
t∗
Ψ=
V
,
V∗
(Ex , Ey , Ez ) =
with E∗ = 0.1 V∗ ℓ−1
∗ and
Ex = −cv
∂Ψ
,
∂x
Ey = −cv
∂Ψ
,
∂y
cv =
V∗
.
ℓ∗ E∗
E
E∗
62
In correspondence to [28] and [5], we perform a coordinate transformation for k
according to
√ ∗
( √
)
√
2m kB TL √
k=
w(1 + αK w) µ, 1 − µ2 cos φ, 1 − µ2 sin φ ,
~
(15)
ε
, the
kB TL
cosine of the polar angle µ and the azimuth angle φ with αK = kB TL α. The main
advantage of the generalized spherical coordinates (15) is the easy treatment of the
Dirac distribution in the kernel (14) of the collision term. In fact, this procedure enables
us to transform the integral operator (4) with the not regular kernel S into an integraldifference operator, as shown in the following.
where the new independent variables are the dimensionless energy w =
We are interested in studying two-dimensional problems in real space; this requires
the full three-dimensional k-space. Therefore, it is useful to consider the new unknown
function Φ related to the electron distribution function via
Φ(t, x, y, w, µ, φ) = s(w)f (t, x, k)|
√
t=t∗ t , x=ℓ∗ (x,y,z) , k=
2m∗ kB TL
~
√
w(1+αK w) ...
,
where
s(w) =
√
w(1 + αK w)(1 + 2αK w),
(16)
is proportional to the Jacobian of the change of variables (15) and, apart from a
dimensional constant factor, to the density of states. This allows us to write the free
streaming operator of the dimensionless Boltzmann equation in a conservative form,
which is appropriate for applying standard numerical schemes used for hyperbolic
partial differential equations. Due to the symmetry of the problem and of the collision
operator, we have
Φ(t, x, y, w, µ, 2π − φ) = Φ(t, x, y, w, µ, φ) .
(17)
Straightforward but cumbersome calculations end in the following transport equation
for Φ:
∂Φ
∂
∂
∂
∂
∂
+
(g1 Φ) +
(g2 Φ) +
(g3 Φ) +
(g4 Φ) +
(g5 Φ) = C(Φ) . (18)
∂t
∂x
∂y
∂w
∂µ
∂φ
The functions gi (i = 1, 2, .., 5) in the advection terms depend on the independent
variables w, µ, φ as well as on time and position via the electric field. They are given
63
by
g1 (·)
=
g2 (·)
=
g3 (·)
=
g4 (·)
=
g5 (·)
=
√
w(1 + αK w)
cx
,
1 + 2αK w
√
√
1 − µ2 w(1 + αK w) cos φ
,
cx
1 + 2αK w
√
]
√
w(1 + αK w) [
µ Ex (t, x, y) + 1 − µ2 cos φ Ey (t, x, y) ,
− 2ck
1 + 2αK w
√
]
[√
1 − µ2
− ck √
1 − µ2 Ex (t, x, y) − µ cos φ Ey (t, x, y) ,
w(1 + αK w)
sin φ
√
ck √
Ey (t, x, y)
w(1 + αK w) 1 − µ2
µ
√
t∗ 2 kB TL
t∗ qE∗
cx =
and ck = √ ∗
.
∗
ℓ∗
m
2m kB TL
The right hand side of (18) is the integral-difference operator
{ ∫ π
∫ 1
′
C(Φ)(t, x, y, w, µ, φ) = s(w) c0
dφ
dµ′ Φ(t, x, y, w, µ′ , φ′ )
with
∫
+
0
π
dφ′
∫
−1
0
1
−1
dµ′ [c+ Φ(t, x, y, w + γ, µ′ , φ′ ) + c− Φ(t, x, y, w − γ, µ′ , φ′ )]
}
− 2π[c0 s(w) + c+ s(w − γ) + c− s(w + γ)]Φ(t, x, y, w, µ, φ) ,
where
(c0 , c+ , c− ) =
2m∗ t∗ √
2 m∗ kB TL (K0 , (nq + 1)K, nq K) ,
~3
γ=
~ωp
kB TL
are dimensionless parameters. We remark that the δ distributions in the kernel S have
been eliminated which leads to the shifted arguments of Φ. The parameter γ represents
the jump constant corresponding to the quantum of energy ~ωp . We have also taken
into account (17) in the integration with respect to φ′ .
In terms of the new variables the electron density becomes
(√
)3
∫
2 m∗ kB TL
n(t∗ t, ℓ∗ x, ℓ∗ y) =
f (t∗ t, ℓ∗ x, ℓ∗ y, k) dk =
ρ(t, x, y) ,
~
R3
where
∫
ρ(t, x, y) =
+∞
∫
0
∫
1
dw
π
dµ
−1
dφ Φ(t, x, y, w, µ, φ) .
(19)
0
Further hydrodynamical variables are the dimensionless x-component of the velocity
∫ +∞ ∫ 1
∫ π
dw
dµ
dφ g1 (w) Φ(t, x, y, w, µ, φ)
0
−1
0
,
ρ(t, x, t)
64
and the dimensionless energy
∫
+∞
∫
dw
0
∫
1
π
dµ
−1
dφ w Φ(t, x, y, w, µ, φ)
0
ρ(t, x, t)
.
Using the new dimensionless variables, Poisson equation becomes
(
)
(
)
∂
∂Ψ
∂Ψ
∂
εr
+
εr
= cp [ρ(t, x, t) − ND (x, y)]
∂x
∂x
∂y
∂y
(20)
with
(√
(√
)−3
)3
2 m∗ k B T L
2 m∗ kB TL ℓ2∗ q
ND (x, y) =
ND (ℓ∗ x, ℓ∗ y) and cp =
.
~
~
ε0
To solve the dimensionless Boltzmann-Poisson system, we need the initial conditions
for Φ and boundary conditions both for Φ and Ψ. These depend on the geometry of the
device and on the problem. In the (w, µ, φ)-space, no boundary condition is necessary.
In fact,
• at w = 0, g3 = 0. At w = wmax , Φ is assumed machine zero;
• at µ = ±1, g4 = 0;
• at φ = 0, π, g5 = 0,
so at those boundaries, the numerical flux always vanishes, hence no ghost point is
necessary for the DG method.
S⃗eMA Journal
no 54(2011), 65–90
ON THE NUMERICAL COMPUTATION OF MOUNTAIN PASS SOLUTIONS
TO SOME PERTURBED SEMI-LINEAR ELLIPTIC PROBLEM
L. MONTORO
Dipartimento di Matematica
Università degli Studi della Calabria
[email protected]
Abstract
In this paper we want to visualize and construct the shape of least energy
solutions to a singularly perturbed problem (M̃ε ) with mixed Dirichlet and
Neumann boundary conditions. Such type of problem arises in several situations as
reaction-diffusion systems, nonlinear heat conduction and also as limit of reactiondiffusion systems coming from chemotaxis. We are mainly interested in using an
appropriate numerical method to show the location and the shape of such type
of solutions when the singular perturbation parameter goes to zero, analyzing the
geometrical effect of the curved boundary of the domain.
A
Introduction
In this short survey we are interested in showing the main results and in analyzing the
main steps of the construction of the shape of positive least-energy solutions (Mountain
Pass Type Solutions) for a mixed perturbed semilinear Dirichlet-Neumann problem as

2
p

in Ω;
−ε ∆u + u = u
∂u
(M̃ε )
u = 0 on ∂D Ω;
∂ν = 0 on ∂N Ω;


u>0
in Ω,
(
)
n+2
where Ω is a smooth bounded subset of Rn , p ∈ 1, n−2
if n ≥ 3 and p ∈ (1, +∞)
if n = 2, ε > 0 is a small parameter, and ∂N Ω, ∂D Ω are two subsets of the boundary
of Ω such that the union of their closures coincides with the whole ∂Ω.
Received: May 14, 2010. Accepted: December 01, 2010.
The author would like to thank warmly Prof. Jesus Azorero, Prof. Andrea Malchiodi and Prof.
Ireneo Peral for many fruitful conversations about the topic and express his gratitude to Departamento
de Matemáticas de la Universidad Autónoma de Madrid for the homely hospitality. The author has been
supported by the Project SB2009-0033, MEC, Spain.
65
66
L. Montoro
These stationary mixed boundary problems appear in different situations and
generally the Dirichlet condition is equivalent to impose some state on the physical
parameter represented by u while instead the Neumann condition gives a meaning at
the flux parameter crossing ∂N Ω, see the introduction of [11, 12, 23] for more specific
comments.
Here below, for completeness and for the reader convenience there are some
classical physical situations where such a type of problem (M̃ε ) appears:
(i) Population dynamics. Assume that a species lives in a bounded region Ω such
that the boundary has two parts, ∂N Ω and ∂D Ω, the first one is an obstacle that
blocks the pass across, while the second one is killing zone for the population.
(ii) Nonlinear heat conduction. In this case (M̃ε ) models the heat (for small
conductivity) in the presence of a nonlinear source in the interior of the domain,
with combined isothermal and isolated regions at the boundary.
(iii) Reaction diffusion with semi-permeable boundary. In this framework we have
that the meaning of the Neumann part, ∂N Ω, is an obstacle to the flux of the
matter, while the Dirichlet part, ∂D Ω, stands for a semipermeable region that
allows the outwards transit of the matter produced in the interior of the cell Ω by
a nonlinear reaction.
Singularly perturbed problems with Neumann or Dirichlet boundary conditions, that is

2
p

in Ω;
−ε ∆u + u = u
∂u
(Nε )
=
0
on ∂N Ω;
∂ν


u>0
in Ω,

2
p

in Ω;
−ε ∆u + u = u
(Dε )
u=0
on ∂D Ω;


u>0
in Ω,
are been studied by many authors, see for example [19], [20], [21] and [22]. Indeed
there is a wide literature regarding the study of this type of problems and we refer also
to the list of references [9], [10], [13], [14], [15], [16], [17], [18], [28] and [29].
The typical concentration behavior of solutions uε to the above(two problems
(Nε )
)
and (Dε ), is via a scaling of the variables in the form uε (x) ∼ U
x−Q
ε
, where Q is
some point of Ω, and where U is a solution of
−∆U + U = U p
in Rn
(or in Rn+ = {(x1 , . . . , xn ) ∈ Rn : xn > 0}),
(1)
the domain depending on whether Q lies in the interior of Ω or at the boundary; in
n+2
(and indeed only if
the latter case Neumann conditions are imposed. When p < n−2
Computation of mountain pass solutions to some semi-linear elliptic problem
67
this inequality is satisfied), problem (1) admits positive radial solutions which decay to
zero at infinity, see [2]:
lim er r
r→+∞
n−1
2
U (r) = αn,p .
Solutions of (M̃ε ) which inherit this profile are called spike layers, since they are highly
concentrated near some point of Ω.
W.M.Ni and I.Takagi in [20] proved that least energy solutions of (Nε ), that is
Mountain Pass solutions, have only one local maximum over Ω achieved at a point
that must lie on the boundary and in a subsequent paper [21] they showed that
the maximum, indeed unique, of the least energy solutions concentrates near global
maxima of the mean curvature H at the boundary.
The main results can be summarized as follows:
Theorem 1 ([20], [21]) For ε sufficiently small the problem (Nε ) possesses a leastenergy solution uε , i.e. a Mountain Pass solution with the following properties:
(i) uε has a unique local (thus global) maximum point Pε ∈ Ω. Furthermore Pε
must lie on the boundary ∂Ω;
(ii) uε → 0 everywhere in Ω outside a small neighborhood of Pε and
uε (Pε ) → U (0), with U is the unique solution of the problem (1) in Rn ;
(iii) H(Pε ) → max H(P ) as ε → 0, where H(P ) denotes the mean curvature of
P ∈∂Ω
∂Ω at P .
Concerning instead (Dε ), W.M.Ni and J.Wei in [22] proved that least energy
solutions have only one local maximum over Ω achieved at the most centered point of
Ω when the perturbation parameter goes to zero.
The main result states:
Theorem 2 ([22]) Let uε be a least-energy solution (i.e. Mountain Pass solution) to
the problem (Dε ). Then, for ε sufficiently small, we have:
(i) uε has at most one local maximum and it is achieved at exactly one point Pε in
1
Ω. Moreover, uε (·+Pε ) → 0 in Cloc
(Ω−Pε \{0}), where Ω−Pε = {x−Pε |x ∈
Ω};
(ii) d(Pε , ∂Ω) → max d(P, ∂Ω) as ε → 0.
P ∈Ω
68
L. Montoro
The mixed case has been analyzed in the two recent works [11, 12]. This case is
more involved because there exists an interface which separates the Dirichlet and the
Neumann parts, namely
IΩ := ∂D Ω ∩ ∂N Ω ̸= ∅.
Here below we give the main results proved in the two works that show the
behaviour of the solutions of (M̃ε ) when the perturbation parameter ε goes to zero.
In particular in the first one [11], the authors determine the geometric conditions to
have some sequence of solutions {uε } (spike-layers type) which approach IΩ when ε
tends to zero. Below we state the main theorem proved:
Theorem 3 ([11]) Suppose Ω ⊆ Rn , n ≥ 2, is a smooth bounded domain, and that
n+2
1 < p < n−2
(1 < p < +∞ if n = 2). Suppose ∂D Ω, ∂N Ω are disjoint open sets of
∂Ω such that the union of the closures is the whole boundary of Ω and such that their
intersection IΩ is an embedded hypersurface. Suppose Q ∈ IΩ is such that H|IΩ is
critical and non degenerate at Q, and that ∇H ̸= 0 points toward ∂D Ω. Then for
ε > 0 sufficiently small problem (M̃ε ) admits a solution uε concentrating at Q, with a
unique maximum point in ∂N Ω, at distance of order ε| log ε| from IΩ .
Remark 1 The non-degeneracy condition on Q can be relaxed requiring that either Q
is a strict local maximum or minimum for H|IΩ or that the local degree of ∇H|IΩ is
non zero for any connected (small) neighborhood of Q in IΩ .
In the second one [12], the authors analyze mainly the behavior of the least energy
solutions to the problem (M̃ε ) under generic assumptions on the domain and on the
interface.
It was shown that Mountain Pass solutions are in fact least energy solutions and
that, given a family of least energy solutions {uε }, their points of maximum must lie
on the boundary of the domain Ω, as in the Neumann case. Moreover, as for (Nε ), the
least energy solutions concentrate at boundary points in the closure of ∂N Ω where the
mean curvature is maximal. We state the main theorems proved:
Theorem 4 ([12]) Suppose Ω ⊆ Rn , n ≥ 2, is a smooth bounded domain, and that
n+2
1 < p < n−2
(1 < p < +∞ if n = 2). Suppose ∂D Ω, ∂N Ω are disjoint open sets
of ∂Ω such that the union of the closures is the whole boundary of Ω and such that
their intersection IΩ is an embedded hypersurface. Then, as ε → 0, the least energy
solution of (M̃ε ) have a unique maximum point which converges to Q ∈ ∂N Ω such
that H(Q) = max H.
∂N Ω
Theorem 5 ([12]) Suppose we are in the situation of Theorem 4. Assume also that
max H is attained only at Q ∈ IΩ , where Q is an isolated maximum point of H|IΩ
∂N Ω
for which ∇H(Q) ̸= 0 points toward ∂D Ω. Then the least energy solution of (M̃ε )
69
behaves as in Theorem 3. If instead max H is attained both at IΩ and at an isolated
∂N Ω
interior point Q̌ of ∂N Ω, and if Q is as in the previous case, the maxima of least energy
solutions to (M̃ε ) converge to Q̌.
In [23] via a useful diffeomorphism y = Ψ(x) (x = Φ(y) with Ψ ≡ Φ−1 ,
straightening the domain around some point belonging to the boundary), we prove
Theorem 6 below that characterizes the shape of the least energy solutions to the
problem (M̃ε ) as the perturbation parameter ε goes to zero. Here we denote Ωε the
scaled domain (after flatten and scaling around Pε = Ψ−1 (Qε ), with Pε belonging to
the interior of Neumann boundary part) and we suppose Qε to be the origin, that is
Qε = 0.
Theorem 6 ([23]) Let uε be a least-energy solution to the problem (M̃ε ) and Ω ⊆
Rn , n ≥ 2, a bounded smooth domain. Let K ⊂ Ωε be a compact containing the
origin and Br+ ⊂ Ωε the set {z ∈ Br , zn > 0}. Moreover let y = Ψ(x) be the
diffeomorphism that straightens a boundary portion around Pε , where Pε ∈ ∂N Ω is,
for ε small enough, the unique maximum point of uε . Let we denote wε (·) = uε (Φ(ε·))
the scaling of the least-energy solution uε after straightening the boundary. Then for
b ε ⊂ Ω,
any α > 0, there exists ε0 such that for ε < ε0 we have two connected sets Ω
e
Ωε ⊂ Ω and ρ positive constant independent of ε, such that
b ε ) < C1 ε,
1. diam(Ω
2. uε (x) − U ( Ψ(x)
ε )
bε)
C 2 (Ω
b ε,
Pε ∈ ∂ Ω
< C εα2 ,
∥wε (z) − U (z)∥C 2 (K) < α,
b ε,
3. |uε (x)| < Cαe−µ1 δ(x)/ε if x ∈ Ω\Ω
e
−µ1 δ(z)
|wε (z)| < Cαe
if z ∈ Ωε \ B r2 ,
(
)1/2
−µ1 ρ/ε
4. ∥uε ∥H 1 (Ω
.
e ε ) ≤ C̃e
D
e
b ε \ ∂Ω) and C1 , C, C̃, µ1 are
where δ(z)
= dist(z, ∂B r2 \ ∂Ωε ), δ(x) = dist(x, ∂ Ω
positive constants depending only on Ω.
For all the details of the proofs and other related results to the problem (M̃ε ) we
recommend to refer to [11, 12, 23] .
In this work we consider the least energy solutions to the problem (M̃ε ), referring
strongly to the numerical analysis’ point of view, giving an approach of their shape
and then presenting the related results. In this type of problems with pattern
formation phenomena, whose main interest is the profiles’ locations of concentration,
the visualization of solutions via computational and numerical methods turns to be very
useful. We refer to the papers [5], [6] and the bibliographies therein for a complete and
accurate description of all these methods.
70
L. Montoro
Since crucial for the application of the numerical algorithm used, we want to
point out that least energy solutions are actually Mountain Pass solutions, see [12,
Proposition 2.3] .
B
The mountain pass shape: a numerical approach.
For our analysis we consider the case n = 2 and p = 3, that is:

2
3

in Ω;
−ε ∆u + u = u
∂u
u = 0 on ∂D Ω;
∂ν = 0 on ∂N Ω;


u>0
in Ω,
(Mε )
where Ω is an elliptical domain of R2 . We want to point out that in the two dimensional
case the interface IΩ is represented by points belonging to R2 . Figure 1 shows how
Dirichlet and Neumann conditions are imposed.
Γ1N
Neu ma nn Condi t i on
Di r i chl et Condi t i on
Γ2D
Γ1D
Dir ichl et Condit ion
Neumann Condit ion
Γ2N
Figure 1: The Elliptical Domain Ω.
In order to construct and aproach the shape of a positive least energy solution of
problem (Mε ) we take advantage of Mountain Pass Algorithm due to Y.S. Choi and
P.J. McKenna, appeared in the pioneering work [7], taking into account the intrinsic
differences, see [23].
Here we consider a perturbative problem with mixed boundary conditions and
defined in an elliptical domain contained in R2 . We need to consider the problem in a
domain with non constant mean curvature H because our goal is finding a spike layer
solution concentrating at the interface: in fact the variation of the mean curvature H
is crucial to visualize a least energy solution to the problem (Mε ) concentrating at the
interface IΩ as ε → 0, see Theorem 4.
71
From numerical point of view a curved boundary domain is difficult to mesh in
order to define all the discrete variables that the algorithm requires. In fact, as known,
such a process is simplest if the geometry is rectangular. However when the domain
has a curved boundary the construction of the mesh is more involved requiring large
manpower and we refer to the books [8] and [27] for more details about.
Moreover to get a Mountain Pass solution concentrating at the interface, we impose
mixed conditions on four disjoint subsets of the boundary ∂Ω, as shown in Figure 1,
in such a way that Dirichlet conditions are imposed on the two boundary parts, Γ1D
and Γ2D , where the curvature H reaches its maximum value. This requires a further
commitment in the processing of the domain geometry and in the code development.
Imposing Dirichlet conditions on Γ1D and Γ2D we avoid that least energy solutions
concentrate where the mean curvature is maximal, see [12]. In this way we get
solutions to (Mε ) concentrating at the interface IΩ according to Theorem 4.
B.1 The Mountain Pass algorithm
First of all, to apply the algorithm, we need to show that the energy functional
associated to the problem (M̃ε ) has the Muontain Pass geometry and satisfies the
Palais-Smale condition. This readily follows by arguments nowadays well understood,
see [3, 4] for example.
In fact, solutions of (Mε ), correspond to critical points of the functional
∫
∫
1
1
Iε (u) =
ε2 |∇u|2 + u2 −
u4 ;
u∈X
2 Ω
4 Ω
(2)
where
1
(Ω) : u ̸≡ 0 and u ≥ 0 in Ω}
X = {u ∈ HD
1
and HD
(Ω) stands for the space of functions in H 1 (Ω) which have zero trace on ∂D Ω,
i.e.
1
HD
(Ω) = {u ∈ H 1 (Ω) : u|∂D Ω = 0}.
Then, it is easy to show, that the functional Iε (see (2)) associated to (Mε ) satisfies
the Mountain Pass Theorem. Indeed by Poincaré inequality we can observe that
Remark 2 For ε fixed,
(∫
∥u∥HD1 =
ε |∇u|
2
2
) 12
Ω
is an equivalent norm to
(∫
∥u∥HD1 =
for the boundary problem (Mε ).
ε |∇u| + u
2
Ω
2
2
) 21
72
L. Montoro
Then by standard arguments Iε has the Mountain Pass geometry. Moreover since the
exponent p is subcritical (and actually since we are in dimension n = 2), a PalaisSmale condition holds true for the functional Iε .
Now we apply to the problem (Mε ) the Mountain Pass algorithm, due to Y.S. Choi
and P.J. McKenna, that is a constructive form of the Mountain-Pass Theorem:
First, we initialize a path from the local minimum w to the point of lower altitude e.
Second, we find the maximum of Iε along this path. Third, we deform the path in such
a way as to make the maximum along the path decrease as fast as possible. Finally,
in the last step we decide whether to stop, if the maximum turns out to have been a
critical point, or repeat the process again.
We outline and rewrite, for completeness and a better understanding, the main steps
in finding an approximate critical point, but for more ideas about the Mountain Pass
algorithm and its implementation, we refer to [6] and [7].
Step 1 of the algorithm: Since for our problem (Mε ) the Mountain Pass geometry
holds, there exist the local minimum w and the point of lower altitude e. We prescribe
a fixed number N , which determine the number of linear segments in the piecewise
linear path Γ0 . One possible starting path would be simply to take a straight line path
from e, to w. It is clear that the initial point e was not optimal. Thus we are led to
make an improvement in the initial path by choosing
g(i) = w +
i
(e − w),
N
1≤i≤N
and then evaluating Iε (g(i)) for i ≥ 1 until we find at i = i0 that
Iε (g(i)) ≤ Iε (w).
Having found such an i0 , we replace the original e with g(i0 ), shortening the distance
between w and e. We now choose our initial Γ0 to be the straight line between w and
e, and the corresponding g(i) to be the equally spaced points along this path.
Step 2 of the algorithm: With a given discretized path g(i), the value of i = im
at which a maximum of Iε (g(i)) occurs is computed. If the number of segments N is
large enough, then 1 ≤ im ≤ N −1, because w is a local minimum and Iε (e) ≤ Iε (w).
This first estimate of the maximum along the path is a little crude. We try to refine it by
the procedures below. Since the path is a discretized one, we will try to locate a higher
maximum between i = im−1 and i = im+1 . Now, we join g(im ) and g(im+1 ) by a
straight line path. In theory we seek α ≥ 0 such that I(g(im ) + αz) stops increasing,
where z = g(im+1 ) − g(im ). In practice, we test successively
g1 =
1
(g(im ) + g(im+1 )),
2
g2 =
1
1
(g(im ) + g(i1 )) g3 = (g(im ) + g(i2 ))
2
2
etc. until either
(a) gi is found such that I(gi ) > I(g(im )) or,
73
(b) ||gi − g(im )||HD1 = (1/2i )||gim+1 − g(im )||HD1 is smaller than a prescribed
tolerance.
In case (a), we evaluate gmid = 12 (g(im )+gi ), and perform a quadratic polynomial
interpolation to find a better value for the maximum location of Iε . In case (b), we do
not change g(im ).
Having checked whether we can improve the estimate of the maximum in the
direction of g(im+1 ), we now perform a similar task with g(im−1 ) and our revised
location of g(im ), so that g(im ) will be moved to attain a higher maximum. We now
refine the path in the neighborhood of the local maximum, by moving some of the
nearby nodes closer. If the H 1 norm distance between g(im ) and g(im+1 ) is larger
than a prescribed value, we then move a prescribed number of mesh points (around
this local maximum) to avoid the missing of saddle points.
Step 3 of the algorithm: The steepest descent direction at the local maximum
location is computed by finding a solution of the problem (Pv ) via finite element
method, solving equation (9), below. Thus, a numerical approximation of 2λv =
v − ε2 u can be computed.
If the difference on the right-hand side is smaller than a prescribed tolerance in the
H 1 norm (see Proposition 7 and (11)), we stop the algorithm and find an approximate
critical point of the functional Iε . Otherwise, up to a prescribed maximum, the
maximum point is moved downhill for a certain descent distance. This distance is
determined by a similar procedure as in step 2. If this distance or the difference in
Iε before and after the descent is smaller than a prescribed tolerance, then we have a
numerical approximation of a critical point and stop the algorithm. If not, repeat steps
2 and 3.
Remark 3 All the algorithm was implemented in a MATLAB code, using different
topics of programming language and different tools of standard numerical analysis.
For all that, we refer to the books [24] and [25].
B.2 The steepest descent direction
1
In our case, the steepest descent direction at u ∈ HD
(Ω) corresponds to finding the
1
direction v ∈ HD (Ω) with ||v||HD1 = 1 that minimize the Fréchet derivative Iε′ (u)v
subject to the constraint that ||v||HD1 = 1.
Introducing the Lagrange multiplier λ, we look for the unconstrained minimum of
1
the functional Jε,u : HD
(Ω) → R defined as:
∫
ε2 ∇u∇v + uv − f (u)v + λ|∇v|2
Jε,u (v) :=
Ω
1
where f (u) = u3 and u, v ∈ HD
(Ω).
(3)
74
L. Montoro
Thanks to Remark 2,
∫ it turns out to be more simple to deal with (3): in fact we can
|∇v|2 = 1 since, for ε fixed, ||∇v||L2 (Ω) is an equivalent
simply minimize over
Ω
norm to ∥v∥HD1 .
1
Now we find the direction v ∈ HD
(Ω) that minimizes the Fréchet derivative of
functional Jε,u defined in (3). If such a function v is a minimum of the functional Jε,u
(3), it must satisfy
′
Jε,u
(v) = 0
i.e.
∫
∫
ε2 ∇u∇w + uw − f (u)w = −
2λ∇v∇w.
(4)
Ω
Ω
Denoting v as:
v := 2λv + ε2 u.
(5)
l(u) := f (u) − u,
(6)
and defining the functional l(·) as
we can arrange equation (4) as:
∫
∫
∇w∇v =
Ω
l(u)w,
Ω
1
Then v ∈ HD
(Ω) corresponds to the unique weak solution (by Lax-Milgram lemma)
for the linear problem:


in Ω;
−∆v = l(u)
∂v
(Pv )
∂ν = 0 on ∂N Ω; v = 0 on ∂D Ω;


v>0
in Ω.
Then by (5) we get the direction v. Moreover |λ| is determined so that ||v||HD1 = 1.
Since
1
Jε,u (v + u) = Jε,u (v) + dJε,u (v)[u] + d2 Jε,u (v)[u, u] + o(||u||3 )
2
∫
and
|∇u|2 .
2λ
Ω
we get that v is a minimum (the steepest descent direction) if λ > 0.
B.3 Numerical approximation of problem (Pv )
In order to solve problem (Pv ) we need to find a numerical approximation of the
solution v, with l(u) known at the mesh points. We use the finite element method
(FEM for short) which is the most appropriate for approximating the solutions of
75
second-order problem posed in variational form over a given functional space V . A
well-known approach to solve such problems is Galerkin’s method, which consists in
defining similar problems, called discrete problems, over a finite-dimensional subspace
Vh of the space V . The finite element method is a Galerkin’s method characterized by
three basic aspects in the construction of the space Vh :
1. a triangulation Th is established over the set Ω, that is, the set Ω is written as a
finite union of finite elements K ∈ Th ;
2. the function vh ∈ Vh are piecewise polynomials, i.e., the spaces PK = {vh|K :
vh ∈ Vh } consist of polynomials;
3. there should exist a basis in the space Vh whose functions have a compact
support.
For more details about such type of method we refer to the books [8, 26, 27].
To solve problem (Pv ), we consider the linear abstract variational problem:
Find u ∈ V such that
∀v ∈ V,
a(u, v) = F (v),
(7)
where the space V , the bilinear form a(·, ·) and the linear form F satisfy the
assumptions of the Lax-Milgram lemma. In our case
1
(Ω),
(i) V = HD
∫
(ii) a(u, v) =
∇u∇v,
∫
(iii) F (v) =
Ω
lv,
Ω
where l is defined in (6).
Then we have to construct a similar problem in finite-dimensional subspace of the
space V . With any Vh of V we associate the discrete problem:
Find uh ∈ Vh such that
∀vh ∈ Vh ,
a(uh , vh ) = F (vh ).
(8)
By Lax-Milgram lemma, we infer that such a problem has only one solution uh . We
denote Nh the number of all mesh points of the domain. We use a piecewise linear
basis functions φj satisfying
{
0 i ̸= j,
φj (Ni ) = δij =
1 i = j,
76
L. Montoro
where Ni is a generic mesh point and i, j = 1, 2, . . . , Nh . Hence we express the
discrete solution uh as
uh (x) =
Nh
∑
uj φj (x),
∀x ∈ Ω,
uj = uh (Nj ),
∀x ∈ Ω,
vi = vh (Ni ).
j=1
and vh ∈ Vh as
vh (x) =
Nh
∑
vi φi (x),
i=1
Then we have to find the coefficients uj such that they are solutions of the linear
system
Nh
∑
a(φj , φi )uj = F (φi ),
i = 1, 2, . . . , Nh .
j=1
Then we get
Au = b
(9)
where A is a Nh × Nh matrix given by
A = [aij ],
∫
∇φj ∇φi ,
aij =
with
Ω
and u and b are vectors defined as
u = [uj ] with
∫
uj = uh (Nj ),
b = [bi ] with bi =
lφi .
Ω
All the integrals in the above expressions are been approximated by Gaussian 2-d
quadrature rule and 2-d composite trapezoidal rule.
Proposition 7 Let uh be the discrete solution to the problem (8). Then the H 1 norm
is given numerically by
1
||uh ||H 1 = (uT Au) 2 ,
where A and u are as in equation (9).
Proof. By a straight calculation, one has that:
∫
∇uh ∇uh =
Ω


Nh
Nh ∑
Nh
Nh
∑
∑
∑


a
=
a(uj φj , ui φi )
uj φj ,
u i φi
j=1
Nh ∑
Nh
∑
j=1 i=1
j=1 i=1
i=1
ui a(φj , φi )uj
=
Nh ∑
Nh
∑
j=1 i=1
= uT Au.
ui aij uj
77
Proposition 7 it is fundamental, through all the algorithm, in finding the H 1 norm of
approximated functions, solutions, and also the H 1 distance between any two given
functions.
A fundamental step in the FEM method is to define the stiffness matrix A
associated to the discretized problem. In our case we perform a mesh via triangular
elements and we use linear piecewise polynomials functions wh over each of such
elements, that is we choose wh = a+bx+dy linear inside each triangle and continuous
across each edge. Then within each triangle, the three coefficients (a, b, d) are uniquely
determined by the values of wh at the three vertices.
Let us denote by (pix , piy ), with i = 1, 2, 3, the coordinates of the three vertices of
each triangle in the cartesian coordinate system. Define the matrix Q as


1 p1x p1y
Q :=  1 p2x p2y  .
1 p3x p3y
Obviously the area ael of each triangle is given by
ael =
1
| det(Q)|.
2
Then denote the matrix c as
c = [cij ] := Q−1 .
The 3 × 3 stiffness element matrix Ael associated to each triangular element K, in
our case is given by

c221 + c231


Ael = ael 
 c22 c21 + c32 c31

c23 c21 + c33 c31
c21 c22 + c31 c32
c222 + c232
c23 c22 + c33 c32
The 3 × 1 element vector b for each K is defined as


h(p1x , p1y )



ael 
 h(p2x , p2y )  .
bel =


3 

h(p3x , p3y )
c21 c23 + c31 c33



c22 c23 + c32 c33 
.

c223 + c233
(10)
Then by summation over all Nh elements we obtain the assembly of the global
stiffness matrix A, that means that the matrix elements are added together, after being
78
L. Montoro
placed in the right position in the global array. In our problem, since the boundary
is curved we need to construct, in a useful way, a table containing the geometric and
topological properties of the mesh. Then we pass from a local problem, defined in each
element K, to the global one defined in Ω, getting the global stiffness matrix A.
For example, here below in Table 1 and Figure 2, we show a good assembly scheme
used in order to construct the stiffness matrix A. For more details about the assembly
scheme we refer to the book [26].
Table 1: Assembly scheme.
Element
1
2
3
4
5
..
.
1
6
7
7
632
722
..
.
Nodes
2
3
1
15
6
2
6
15
650 614
714 713
..
..
.
.
1
1
15
6
614
3
15
2
2
4
7
632
650
Figure 2: Number of elements and nodes.
Remark 4 All the integrals in the above expressions are been approximated by
Gaussian 2-d quadrature rule and 2-d composite trapezoidal rule. Denoting K a
triangular element, we can compute the integral of a function g(x) defined in K as
∫
area(K) ∑
g(pix , piy ).
3
i=1
3
g(x)dx =
K
79
In this way, for example, we get (10).
Another important goal is to create a mesh for the domain Ω which is elliptical.
With respect to the domains considered in [7], in our case the main difficulty is that
the boundary is curved. In general a mesh requires a choice of mesh points (vertex
Figure 3: Subdivision of the domain Ω into triangles.
nodes) and a triangulation, see Figure 3. The FEM triangulation of a domain with
a curved boundary is not a trivial task, see [6], because the coding work increases a
lot since we need a mesh refinement to better approximate the curved boundary. The
mesh should have small elements to resolve the fine details of the geometry, but larger
size, where possible, to reduce the total number of nodes. We have chosen triangular
elements because triangles are obviously better at approximating a curved boundary.
We subdivide the basic region Ω into triangles. The union of these triangles will be
a polygon Ωh . In general, if ∂Ω is a curved boundary, there will be a nonempty skin
Ω \ Ωh . We assume that Ωh is a subset of Ω and that no vertex of any triangles lies
along the edge of another one.
C
Graphic results: the spike-layer solution in the mixed case
Here below we present the graphic results obtained applying the numerical method
described in the previous section. The algorithm was implemented using a MATLAB
code. A detailed description of the code of all procedures of the algorithm is included
in [1].
Below, in the different figures, we analyze the behavior of the least energy solution
when the perturbation parameter ε goes to zero. We run the algorithm fixing, every
time, a smaller different ε-value.
80
L. Montoro
We denote dn the absolute convergence error defined here as
{
}
dn = max ||unε − un−1
||HD1 (Ω) , |Iε (unε ) − Iε (un−1
)| ,
ε
ε
(11)
where unε ( resp. un−1
) is the approximate least energy solution at the Step n (Step
ε
n−1) of the algorithm. The process stops when a fixed desired dn indicator is achieved.
In this work, we set dn = 10−3 .
As Figure 1 shows, we have considered an elliptical domain with boundary
subdivided in four parts, that is, we set:
∂N Ω =
Γ1N ∪ Γ2N ,
∂D Ω =
Γ1D ∪ Γ2D ,
with Γ1N ∩ Γ2N = ∅, Γ1D ∩ Γ2D = ∅ and ∂Ω = ∂N Ω ∪ ∂D Ω.
Since we are working in a domain Ω ⊂ R2 , the interface IΩ consists in four points
belonging to R2 space. On the two boundary parts, where the curvature reaches its
maximum value we impose Dirichlet conditions, see Figure 1.
In [12], we have showed that the concentration of the limit point Pε cannot occur
on the Dirichlet boundary part. The point Pε is such that
uε (Pε ) = maxuε (P ),
x∈Ω
and up to a subsequence, we suppose that Pε → P0 . In fact if
dist(Pε , ∂D Ω)
→ ∞,
ε
x
the least energy
ε
solution uε would converge to the ground state solution U of problem (1) in Rn in C 2
sense, with a critical level
cε = εn {I(U ) + o(1)}
as ε → 0, see [11, Corollary 3.7], up to a scaling of the type x′ =
bigger than the least energy solution critical level
cε = εn {
I(U )
+ o(1)},
2
which is a contradiction (see [12]). Then, since the two points where the curvature
maximizes belong to Γ1D and Γ2D , the maximum point Pε should concentrate at a point
P0 ∈ IΩ , see Theorem 4.
For each ε-value considered we run the algorithm getting a three-dimensional view,
see for example Figure 4, and a two-dimensional view, see for example Figure 5.
81
The three-dimensional view shows better the profile of such a solution: in fact
that is called spike layer, since it is highly concentrated near some point of Ω. As
the perturbation parameter ε becomes smaller, see Figure 6, Figure 8 and Figure 10,
the Mountain Pass type solutions uε tend to zero uniformly on every compact set of
Ω \ {P0 } (see Theorem 6), while there exist δ ≥ 1 and Pε such that uε (Pε ) ≥ δ. This
is evident utilizing the equation
−ε2 ∆u + u = u3
when Pε is in the interior and otherwise, when Pε ∈ ∂N Ω this follows from the
boundary Hopf lemma.
Instead, in the two dimensional view, see Figure 7, Figure 9 and Figure 11, we note
that the maximum point Pε belongs to the Neumann boundary part and, as ε goes to
zero, this point reaches the interface IΩ .
Moreover the support of the least energy solution becomes smaller when ε goes
to zero. In fact, for example, giving a look at Figure 10 and Figure 11, we note that
the maximum point Pε concentrates at the interface IΩ and the spike layer’s profile is
concentrated around the maximum point in a small support, as Theorem 6 states.
82
L. Montoro
Figure 4: Three-dimensional view with ε = 0.7 and max uε = 2.14.
Figure 5: Two-dimensional view with ε = 0.7.
83
84
L. Montoro
85
86
C.1
L. Montoro
A concluding case: the spike-layer solution in the Dirichlet case
As last application, we show as the Mountain Pass algorithm can be applied to the
singularly perturbed Dirichlet problem

2
3

in Ω;
−ε ∆u + u = u
(12)
u=0
on ∂Ω;


u>0
in Ω.
In fact, also in this case, the least energy solutions come from the Mountain Pass
Theorem. While in the Mixed case (M̃ε ) the spike layer solution with its unique
maximum concentrates in the closure of the Neumann boundary part ∂N Ω where the
curvature maximize, in the Dirichlet case (Dε ) the spike layer solution has one local
maximum achieved at a point that maximize the distance from the boundary ∂Ω, see
Theorem 2. We can use the Mountain Pass algorithm to treat the perturbed problem
(12) over the same elliptical domain, making the necessary changes with respect to
the mixed case. As Figures 12, 13, 14 and 15 show, we get a spike-layer solution
concentrating, as ε goes to zero, at the center of symmetry of the elliptical domain in a
support becoming smaller around the unique maximum point Pε .
87
88
L. Montoro
References
[1] Files are available online at http://www.mat.unical.it/∼ montoro
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S⃗eMA JOURNAL BEST PAPER AWARD
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development of applied mathematics and its growing influence in the everyday life
of modern societies and also aware of the need to encourage the transfer of scientific
knowledge within our community, is proud to announce the “S⃗eMA Journal Best Paper
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dissemination of applied mathematics, emphasizing its relevance and potential. There
is an increasingly large body of applied interests in mathematics and the guidelines
below seek to enforce those subjects that have traditionally been in the core of the
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and computational techniques in mathematical modelling in science.
GENERAL RULES
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“S⃗eMA Journal Best-Paper Award” on a yearly basis.
2. All papers published in the volumes of the S⃗eMA Journal of the corresponding
year will be eligible for the prize.
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Editor-in-Chief of the S⃗eMA Journal, who will be the chair, the President of
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If any of the authors is not a member of S⃗eMA, he or she will be automatically
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91

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