Surrey: 14th November 2008 - Heriot

Transcription

Grassmannian spectral shooting and the stability
of multi-dimensional travelling waves
Simon J.A. Malham
Veerle Ledoux (Ghent),
Jitse Niesen (Leeds),
Vera Thümmler (D-fine)
Heriot–Watt University, Edinburgh, UK
Surrey: 14th November 2008
Grassmannian spectral shooting and the stability of multi-dimensional travelling waves
Spectral problems
Parabolic nonlinear systems on R × T:
∂t U = B ∆U + c ∂x U + F (U),
Travelling wave Uc . Small perturbations U satisfy:
B ∆U + c ∂x U + DF (Uc )U = λU.
Two main solution approaches:
◮
Projection.
◮
Shooting.
Motivation
Workshop at AIM, Palo Alto, May 2005:
Stability criteria for multi-dimensional waves and
patterns.
Organised by: Chris Jones, Yuri Latushkin, Bob Pego, Björn
Sandstede and Arnd Scheel.
Setup
On R:
B ∆U + c ∂x U + DF (Uc )U = λU
⇔
Y ′ = A(x; λ) Y
Assume for λ ∈ Ω ⊆ C:
◮
Exponential dichotomies on R− and R+ ;
◮
Same Morse index k (Sandstede 2002).
Matching condition (Alexander, Gardner and Jones 1990):
D(λ) ≡ e−
Rx
0
TrA(ξ;λ) dξ
det Y − (x; λ) Y + (x; λ) .
Numerical issues
◮
Computational domain.
◮
Different exponential growth rates.
◮
Polynomial complexity.
◮
Flow singularities?!
◮
Where to match?
◮
Retaining analyticity.
◮
How to project transversely.
Outline
1 Grassmann manifolds and flows
2 Patch evolution (GGEM)
3 Applications (planar fronts)
4 Transverse Fourier projection
5 Application (wrinkled fronts)
6 Future work
Stiefel and Grassmann manifolds
◮
Stiefel manifold:
V(n, k) = {k-frames centred at the origin}.
◮
Grassmann manifold:
Gr(n, k) = {k-dimensional subspaces of Cn }.
◮
Trivial fibre bundle:
V(n, k) ∼
= Gr(n, k) × GL(k)
◮
Projection π : k-frame 7→ spanning k-plane.
Representation
Coordinate patches Ui: multi-index i = {i1 , . . . , ik } ⊂ {1, . . . , n}.
Example: U{1,...,k} uniquely represented by:


1
0
···
0
 0
1
···
0 


 ..
..
.. 
..
 .

.
.
.


 0

0
·
·
·
1

yi◦ = 
ŷk+1,1 ŷk+1,2 · · · ŷk+1,k  .


ŷk+2,1 ŷk+2,2 · · · ŷk+2,k 


 ..
..
.. 
.
.
 .
.
.
. 
ŷn,1
ŷn,2
···
ŷn,k
Local coordinate chart ϕi : Ui → C(n−k)k given by ϕi : yi◦ 7→ ŷ .
Grassmannian flows
Y ′ = A(x, Y ) Y .
Substitute decomposition Y = yi◦ ui:
yi′◦ ui + yi◦ ui′ = (Ai + Ai◦ yi◦ ) ui,
Project onto ith and i◦ th rows:
ŷ ′ = c + d ŷ − ŷ (a + b ŷ )
and
ui′ = (a + b ŷ ) ui,
where a = Ai×i, b = Ai×i◦ , c = Ai◦ ×i and d = Ai◦ ×i◦ .
Linear vector field: A = A(x) only?
Drury–Oja flow
Humpherys and Zumbrun; QR-decomposition of Y ∈ V(n, k):
Q ′ = (In − QQ † )A(x)Q,
(det R)′ = Tr Q † A(x)Q (det R).
Corresponds to Riccati with u ′ = −ŷ † (c + d ŷ ) u.
Grassmannian Gaussian elimination method (GGEM)
ϕ−1
i
C(n−k)k
/ Ui
id
/ V(n, k)
GGEM
C(n−k)k
o
ϕi ′
Ui′ o
QOGE
(ΛY0 )∗
/ GL(n)
log
/ gl(n)
Magnus
RK
V(n, k) o
GL(n) o
gl(n)
ΛY0
exp
Quasi-optimal Gaussian elimination (QOGE)
GE with free stepwise max pivot, generates: Ym+1 = yi◦ L.

∗
∗

∗

∗

∗

∗

∗

.
 ..

∗

∗
∗
∗
∗
∗
∗
∗
∗
..
.
∗
∗
∗
∗
∗
∗
∗
..
.
∗
∗
∗
∗
∗
∗
∗
..
.
···
···
···
···
···
···
···
..
.
∗ ∗ ∗ ···
∗ ∗ ∗ ···
∗ ∗ ∗ ∗ ···


1
∗


∗
∗
0
∗


∗
∗


∗

∗


∗ −→ 
0
∗

∗

.
.. 
 ..
.


∗
∗


0

∗
∗
0
∗
1
∗
∗
0
∗
..
.
0
∗
0
∗
∗
1
∗
..
.
0
∗
0
∗
∗
0
∗
..
.
···
···
···
···
···
···
···
..
.
∗ ∗ ∗ ···
0 0 0 ···
∗ ∗ ∗ ∗ ···

0
∗

0

∗

∗

0

∗

.. 
.

∗

1
∗
Applications (planar fronts)
◮
◮
D(λ) ≡ e−
Rx
0
TrA(ξ;λ) dξ
det Y − (x; λ) Y + (x; λ) .
det Y − Y + = det yi◦−
yi◦+ · det ui− · det ui+ .
◮
D(λ; x∗ ) ≡ det yi◦−
◮
Exponentially scale GGEM.
yi◦+ · det L− · det L+ .
Boussinesq system
PDE: utt = (1 − c 2 ) uxx + 2c uxt − uxxxx − (u 2 )xx .
√
Solitary waves: ū(x) ≡ 32 (1 − c 2 )sech2 21 1 − c 2 x .
Stable when 1/2 < |c| < 1 and unstable when |c| < 1/2.

0
1
0

0
0
1
A(x; λ) = 

0
0
0
−λ2 − 2ū ′′ 2λc − 4ū ′ (1 − c 2 ) − 2ū

0
0
.
1
0
Boussinesq: GGEM
−5
1.5
x 10
1.5
1
1
0.5
0.5
−
−0.5
Y
D(λ)
0
0
−1
−1.5
−0.5
−2
−1
−2.5
−3
0
0.05
0.1
λ
0.15
0.2
−1.5
−8
−6
−4
−2
0
ξ
2
4
6
8
Figure: Evans function with GGEM-RK and x∗ = 8 (left panel). Entries
of yi for λ = 0.15543141 (right panel).
Boussinesq: Evans function vs matching point
CO−RK
Riccati−RK
Mobius−M
GGEM−LG
GGEM−RK
0
10
|D(λ)|
−5
10
−10
10
−8
−6
−4
−2
0
2
4
6
8
matching point
Figure: |D(λ)| for λ equal to the eigenvalue for different matching
points. The number of steps in the equidistant mesh was N = 512.
Boussinesq: error vs matching point
CO−RK
Riccati−RK
Mobius−M
GGEM−LG
GGEM−RK
Riccati−QOGE
error in the eigenvalue
−6
10
−7
10
−8
10
−9
10
−8
−6
−4
−2
0
2
matching point
4
6
8
Figure: Error in the eigenvalue for different choices of the matching
point: N = 512.
Autocatalytic fronts
∂t u = δ∆u + c∂x u − uv m ,
∂t v = ∆v + c∂x v + uv m .

0
0
1
0

0
0
0
1 
,
A(x; λ) = 
m
m−1
λ/δ + v̄ /δ mūv̄
/δ −c/δ 0 
−v̄ m
λ − mūv̄ m−1
0
−c

Autocatalytic fronts: Evans contours
−7
−14
x 10
2.2
0.12
0.12
1.8
0.1
x 10
12
0.12
4.5
2
10
4
0.1
0.1
3.5
1.6
8
0.08
1.2
0.06
2.5
0.06
1
6
0.06
2
0.8
0.04
0.08
3
Im(λ)
1.4
Im(λ)
Im(λ)
0.08
0.04
4
0.04
1.5
0.6
0.02
1
0.02
0.4
0
0.005
0.01
0.015
0.02
Re(λ)
0.025
0.03
0.035
2
0.5
0.2
0
0.02
0
0.04
0
0.005
0.01
0.015
0.02
Re(λ)
0.025
0.03
0.035
0
0.04
0
0.005
0.01
0.015
0.02
Re(λ)
0.025
0.03
0.035
0.04
−7
−14
x 10
2.2
0.12
0.12
1.8
0.1
x 10
12
0.12
4.5
2
10
4
0.1
0.1
3.5
1.6
8
0.08
1.2
0.06
2.5
0.06
1
0.8
0.04
0.08
3
Im(λ)
1.4
Im(λ)
Im(λ)
0.08
6
0.06
2
0.04
1.5
4
0.04
0.6
0.02
0.4
1
0.02
0.2
0
0
0.005
0.01
0.015
0.02
Re(λ)
0.025
0.03
0.035
0.04
0.02
2
0.5
0
0
0.005
0.01
0.015
0.02 0.025
Re(λ)
0.03
0.035
0.04
0
0
0.005
0.01
0.015
0.02
Re(λ)
0.025
0.03
0.035
0.04
Figure: Contour lines of |D(λ)| for δ = 0.1 and m = 9 using the CO-RK
and GGEM-LG (order: top three down to bottom three), matching at
positions x∗ = −8, 0, +8 (left to right).
Autocatalytic fronts: error vs matching point
−5
10
CO−RK
Riccati−RK
Mobius−M
GGEM−LG
−6
10
−7
10
−10
−8
−6
−4
−2
0
2
4
6
8
10
matching point
Figure: Error in the eigenvalue when δ = 0.1 and m = 9: N = 256.
Transverse Fourier basis
On R × T we have:
B∆U + c ∂x U + DF (Uc )U = λU.
On the Fourier modes k = −K , −K + 1, . . . , K :
∂x Ûk = P̂k ,
∂x P̂k = λB −1 Ûk + (k/L̃)2 Ûk − c B −1 P̂k −
K
X
ν=−K
B −1 D̂k−ν Ûν ,
Large ODE system
ON(2K +1)
IN(2K +1)
Û
Û
=
∂x
Ã3 (λ) + Â3 (x) −c B −1 ⊗ I2K +1
P̂
P̂
with Ek (λ) ≡ λB −1 + (k/L̃)2 IN :


E−K (λ)
O
···
O
 O
E−K +1 (λ) · · ·
O 


Ã3 (λ) =  .
.
.. 
.
.
.
.
 .
.
.
. 
O
···
O E+K (λ)


D̂0
D̂−1 · · ·
D̂−2K
 D̂1
D̂0
· · · D̂−2K +1 


Â3 (x) = −B −1 ⊗  .
.
..
..
..

 ..
.
.
.
D̂2K D̂2K −1 · · ·
D̂0
Computing travelling waves: freezing method
Substitute U(x, y , t) = V (x − γ(t), y , t) into original PDE:
∂t V = B ∆V + γ ′ (t)∂x V + F (V ),
Z
T
∂x V̂ (x, y , t)
V̂ (x, y , t) − V (x, y , t) dx dy .
0=
R×T
(Developed by Beyn and Thümmler.)
Wrinkled fronts (cubic autocatalytic system)
Figure: The wrinkled front for δ = 3. Left panel: u component. Right
panel: v component. Cut from domain [−150, 150] × [−60, 60], grid
300 × 240, FE rectangles.
Wrinkled front: Evans function for δ = 2.5
−6
2
0.005
0
x 10
0
−0.005
−2
D(λ)
D(λ)
−0.01
−0.015
−0.02
−4
−6
−0.025
−8
−0.03
−0.035
−8
−6
−4
λ
−2
−10
0
−15
−3
x 10
−10
λ
−5
0
−4
x 10
Wrinkled front: Evans function for δ = 3
−6
2
x 10
1
D(λ)
0
−1
−2
−3
−4
−5
−6
−5
−4
−3
−2
λ
−1
0
1
2
−3
x 10
−9
−8
6
1
x 10
x 10
−8
4
x 10
4
0
D(λ)
D(λ)
D(λ)
2
0
2
−2
−4
−5.45
0
−1
−2
−5.4
−5.35
λ
−5.3
−3
x 10
−2
−8
−6
−4
λ
−2
0
−4
x 10
−4
1.56
1.58
λ
1.6
−3
x 10
Wrinkled front: Eigenvalues for δ = 3
K
3
4
5
6
7
8
9
..
.
24
0.001609
0.001609
0.001589
0.001589
0.001589
0.001589
0.001589
0.001589
0.001592
Eigenvalues (Evans function)
−0.000026 −0.000781 −0.001296
0.000002 −0.000001 −0.000519
0.000002 −0.000001 −0.000519
−0.000002 −0.000003 −0.000515
−0.000002 −0.000003 −0.000515
−0.000002 −0.000003 −0.000515
−0.000002 −0.000003 −0.000515
..
.
−0.000002 −0.000003 −0.000515
Eigenvalues (ARPACK)
0.000000 0.000000 −0.000514
−0.000670
−0.000670
−0.000720
−0.000720
−0.000721
−0.000721
−0.000721
−0.000721
−0.000719
Wrinkled front: contour integration
12
argument (multiples of π)
10
8
6
4
2
0
−2
0.0001
0.0001i
3i
λ
1.5+1.5i
1.5
Figure: Left panel: contour. Right panel: arg D(λ) when λ transverses
the top half. δ = 3.
Future work
◮
Multiple sources of error: relative influence?
◮
Control theory: Lagrangian Grassmannian.
◮
Schubert cells.
Spectral plane
Complex λ plane
15
10
Im[λ]
5
0
−5
−10
−15
−10
−5
0
5
Re[λ]
10
15
Boussinesq: Evans function
−4
−3
x 10
x 10
7
2.5
6
2
5
4
D(λ)
D(λ)
1.5
1
3
2
0.5
1
0
0
−1
−0.5
0
0.05
0.1
λ
0.15
0.2
0
0.05
0.1
λ
0.15
0.2
Figure: Wave speed c = 0.4. Left plot: Riccati-RK (i− = {1, 2} over
[−8, 0] and i+ = {3, 4} over [0, 8]). Right plot: CO-RK.
Boussinesq: eigenvalue error I
CO−RK
Riccati−RK
Mobius−M
GGEM−LG
GGEM−RK
Riccati−QOGE
−4
10
−6
10
−8
10
−10
10
0
1
10
10
time (s)
Figure: Error in the eigenvalue vs cputime, matching at x∗ = 0.
Boussinesq: eigenvalue error II
−4
10
CO−RK
Mobius−M
GGEM−LG
GGEM−RK
Riccati−QOGE
−6
10
−8
10
−10
10
−12
10
0
1
10
10
time (s)
Figure: Error in the eigenvalue vs cputime, matching at x∗ = 8.
Boussinesq: Evans function
2
5
4
1.5
3
2
1
−
0.5
y
D(λ)
1
0
0
−1
−2
−0.5
−3
−1
−1.5
0
−4
0.05
0.1
λ
0.15
0.2
−5
−8
−6
−4
−2
0
ξ
2
4
6
8
Figure: Left: Evans function with Möbius–Magnus: x∗ = +8, Right:
entries in ŷ − when λ = 0.15543141.
Autocatalytic fronts: Evans contours
0.22
0.12
0.12
6
0.12
0.1
5
0.1
0.08
4
0.08
0.06
3
0.04
2
0.04
0.02
1
0.02
1.6
0.2
0.18
0.1
1.4
0.16
0.12
0.06
0.1
0.08
0.04
Im(λ)
1.2
0.14
Im(λ)
Im(λ)
0.08
1
0.06
0.8
0.6
0.06
0.4
0.04
0.02
0.2
0.02
0
0
0.005
0.01
0.015
0.02 0.025
Re(λ)
0.03
0.035
0
0.04
0
0.005
0.01
0.015
0.02 0.025
Re(λ)
0.03
0.035
0
0.04
0
0.005
0.01
0.015
0.02
Re(λ)
0.025
0.03
0.035
0.04
0.22
0.12
0.12
6
0.12
0.1
5
0.1
0.08
4
0.08
0.06
3
0.04
2
0.04
0.02
1
0.02
1.6
0.2
0.18
0.1
1.4
0.16
0.12
0.06
0.1
0.08
0.04
Im(λ)
1.2
0.14
Im(λ)
Im(λ)
0.08
1
0.06
0.8
0.6
0.06
0.04
0.02
0.4
0.2
0.02
0
0
0.005
0.01
0.015
0.02
Re(λ)
0.025
0.03
0.035
0.04
0
0
0.005
0.01
0.015
0.02 0.025
Re(λ)
0.03
0.035
0.04
0
0
0.005
0.01
0.015
0.02
Re(λ)
0.025
0.03
0.035
0.04
Figure: Contour lines of |D(λ)| for δ = 0.1 and m = 9 using the
Riccati-RK, Möbius–Magnus (order: top three down to bottom three),
matching at positions x∗ = −8, 0, +8 (left to right).
Wrinkled front: same with Drury–Oja
−42
2
x 10
1
D(λ)
0
−1
−2
−3
−4
−5
−6
−5
−4
−3
−44
x 10
−1
−43
5
x 10
0
D(λ)
D(λ)
2
−3
−2
−2
1
x 10
x 10
0
−5.45
0
2
2
D(λ)
−2
λ
−45
0
−4
−5.4
−5.35
λ
−5.3
−3
x 10
−8
−6
−4
λ
−2
0
−4
x 10
−5
1.56
1.58
λ
1.6
−3
x 10

Surrey: 14th November 2008 - Heriot

Transcription

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