Upwinding - Universität Stuttgart

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Upwinding
Universität Stuttgart
Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/cover_sheet.tex
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Table of contents
1. Introduction Upwinding (Transport Equation)
2. Comparison with the Central Difference Scheme
3. Upwinding (Advection Equation)
4. Courant number
5. Alternative Interpretation
6. Further discretization schemes
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/toc_9.tex
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Introduction to upwind methods
Before we have mainly considerd centered methods, symmetric
about the point where we are updating the solution.
These methods are appropriate for all systems, in which the
information is travelling uniformly in all directions. Differential
equations that are describing these kinds of problems are called
elliptic equations. The model example is a stone that is thrown
into a lake. The waves are propagating in circles.
The Laplace equation of groundwater flow and the heat diffusion
equation are examples for this kind of physiccal problems.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/intro_upwind_1.tex
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For hyperbolic problems, however, we expect information to
propagate in a certain direction. Supercritical flow in rivers or
supersonic speed of an airplane are physical examples.
In an advection/diffusion problem, e.g. represented by the
transport equation
∂c
∂2c
v
−D 2 =0
∂x
∂x
we speak of a parabolic equation. This is the type of equation
that is lying between the two extremes (elliptic and hyperbolic).
The advective term in the equation represents the hyperbolic
character, whereas the diffusion term represents the elliptic part.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/intro_upwind_1b.tex
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The Peclet number
vL
Pe =
D
describes the relation between the advective and the diffusive
part. If the Peclet number is very small the diffusive part
dominates and central discretization schemes are the best
choice. If, on the ohter hand, the Peclet number is very high,
then the advective term is dominating and Upwinding methods
are appropriate.
Upwinding schemes use backward differences to discretize the
advective term. The information is travelling with the flow and it
therefore makes sense to take only the upstream information
into account.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/intro_upwind_1c.tex
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Comparison of upwinding and central
scheme
In the Upwinding method we consider the upstream and the
current node in the advection term:
ci − ci−1
ci+1 − 2ci + ci−1
=0
−D
v
2
∆x
(∆x)
Central difference scheme:
ci+1 − ci−1
ci+1 − 2ci + ci−1
v
=0
−D
2
2∆x
(∆x)
Take the upwinding scheme and add and subtract the advection
term of the central scheme.
ci − ci−1
ci+1 − ci−1
ci+1 − ci−1
ci+1 − 2ci + ci−1
v
=0
+v
−v
−D
2
∆x
2∆x
2∆x
(∆x)
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/comp_upwind_central.tex
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scheme
ci − ci−1
ci+1 − ci−1
ci+1 − ci−1
ci+1 − 2ci + ci−1
v
−v
+v
−D
∆x
2∆x
2∆x
(∆x)2
{z
}
|
central scheme
ci+1 − ci−1
ci+1 − 2ci + ci−1
2ci − 2ci−1 − ci+1 + ci−1
+v
−D
v
2∆x
2∆x
(∆x)2
−ci+1 + 2ci − ci−1
ci+1 − ci−1
ci+1 − 2ci + ci−1
v
+v
−D
2∆x
2∆x
(∆x)2
v∆x ci+1 − 2ci + ci−1
ci+1 − ci−1
ci+1 − 2ci + ci−1
−
+v
−D
2
2
(∆x)
2∆x
(∆x)2
ci+1 − ci−1
ci+1 − 2ci + ci−1
v∆x D ci+1 − 2ci + ci−1
+v
−D
−
2
D 2
(∆x)
2∆x
(∆x)2
=0
=0
=0
=0
=0
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/comp_upwin_central2.tex
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scheme
ci+1 − ci−1
ci+1 − 2ci + ci−1
P e ci+1 − 2ci + ci−1
+v
=0
−D
− D
2
2
2
(∆x)
2∆x
(∆x)
ci+1 − ci−1
P e ci+1 − 2ci + ci−1
v
=0
−D 1+
2
2∆x
2
∆x
| {z }
The diffusion is artificially increased.
⇓
P e ∂2c
∂c
− D(1 +
) 2 =0
v
∂x
2 ∂x
This diffusion is referred to as numerical diffusion.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/comp_upwin_central3.tex
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Combination of the methods
One can also try to combine the advantages of the two methods.
Use the better accuracy of the central difference scheme with
the stability of the upwind scheme.
The upwind coefficient α is the weighting factor.
Central differences
∂c
(ui+1 − ui−1 )
=
∂x centr
2∆x
Upwinding:
∂c
(ui − ui−1 )
=
∂x upw
∆x
Linear interpolation with upwind-coefficient α
∂c
(ui − ui−1 )
(ui+1 − ui−1 )
=α
+ (1 − α)
∂x adv
∆x
∆x
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/comb_methods.tex
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Advection equation
In a scalar advection problem, e.g. the hyperbolic advection
equation
∂c
∂c
+v
=0
(1)
∂t
∂x
the velocity is independent of the concentration.
The linear, hyperbolic first-order equation has been studied
intensively and several schemes have been proposed to solve
this equation. Here we will present some of these.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/adv_eqn.tex
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Upwinding (Advection equation)
Using an explicit time discretization and a backward difference
scheme in space leads to the first-order upwind method for the
advection equation
1
v
n
n+1
n
n
ui − ui +
ui − ui−1 = 0
∆t
∆x
un+1
i
tn+1
tn
uni−1
uni
uni+1
x
where the chained lines indicate time and space levels and the
solid boxes are the control volumes.
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Upwinding II (Advection equation)
The first-order upwind method for the advection equation is
conditionally stable.
v∆t n
n+1
n
ui = ui −
(ui − uni−1 ).
∆x
The ratio
v∆t
Cr =
∆x
is known as the Courant Number. It can be interpreted as the
physical velocity v over the grid velocity ∆x/∆t.
For Cr = 1 the solution is exact, for smaller Courant numbers
the solution is stable but a large amount of numerical diffusion is
introduced.
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Alternative interpretation
Think of uni as being the value at the grid point, uni ≈ ũ(xi , tn ) in
a finite difference setting. Then since u(x,t) is constant along its
characteristic, i.e. the path line, we expect:
un+1
≈ ũ(xi , tn+1 ) = ũ(xi − v∆t, tn )
i
un+1
i
tn+1
∆t
tn
x
uni−1
uni
xi − v∆t
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/altern_interpr_1.tex
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Alternative interpretation
If un+1
represents the value at a grid point, then we can trace the
i
characteristic back and interpolate. We approximate this value
by a linear interpolation between the grid values uni−1 and uni .
un+1
i
v∆t n
v∆t
uni
=
ui−1 + 1 −
∆x
∆x
−→ This is simply the upwind method.
Here we can see that we have a weighted average between the
two nodes. This makes physical sense as long as the
characteristic really intersects the old time level between the two
nodes, which is again our condition Cr ≤ 1.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/altern_interpr_2.tex
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Central difference scheme
The unstable centered method applied to the advection equation
with an explicit time discretization scheme yields the following
equation
v
1
n+1
n
n
n
ui − ui +
ui+1 − ui−1 = 0.
∆t
2∆x
v∆t n
n+1
n
n
ui = ui −
ui+1 − ui−1 .
2∆x
Note the slight difference in the advection term. The additional
factor 21 and the uni+1 instead of uni .
This solution is always unstable!
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/central_diff.tex
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Lax Scheme I
t
unknown
known
n+1
∆t
n
∆x
∆x
i−1
i
s
i+1
The diffusive (lax) method distinguishes itself from the unstable
model insofar as that for the time derivation not uni is used, but a
n+1
mean from un+1
and
u
i−1
i+1 .
This scheme is stable for Cr ≤ 1.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/lax_scheme_1.tex
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Lax Scheme II
uni+1 − uni−1
∂u
≈
∂x
2∆x
(as in the unstable method)
2un+1
− uni−1 − uni+1
∂u
i
≈
∂t
2∆t
(new compared to
the unstable method)
or, more general:
∂u
≈
∂t
(0 ≤ α ≤ 1;
un+1
i
n
n
n
− αui + (1 − α) ui−1 + ui+1 /2
∆t
for α = 1 results the unstable model).
u ≈ uni
uni−1 + uni+1
or u ≈
2
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/lax_scheme_2.tex
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Preissmann Scheme
t
k+1
Θ ∆t
unknown
known
(1− Θ )∆ t
approximated
k
i
∆ x ij
2
∆ x ij
2
j
s
∆ x ij
The Preissmann scheme differs from the before mentioned
schemes by also averaging the time derivative.
The first derivatives are depictable in the point + in the space
and time direction.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/preismann_1.tex
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Preissmann Scheme II
u+
′x
u+
′t
k+1
uk+1
−
u
ukj − uki
∂u+
j
i
+ (1 − Θ)
,
≈Θ
=
∂x
∆xij
∆xij
!
k+1
k+1
k
k
uj + ui
∂u+
1 uj + ui
=
≈
−
∂t
∆t
2
2
1
k+1
k+1
k
k
=
uj − uj + ui − ui .
2∆t
For the quantity of state in the calculation point, the mean value
is established:
k+1
k+1
k
k
u
+
u
+
u
u
i
j
j
i
+ (1 − Θ)
.
u+ ≈ Θ
2
2
The method is stable for Θ > 0, 5. yields best results between:
Θ ≈ 0, 55 − 0, 6. and creates numeric damping for larger values.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/preismann_2.tex
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Further schemes
Leap-frog
un+1
−un−1
i
i
2∆t
+
n
un
i+1 −ui−1
v 2∆x
= 0.
Lax Wendroff
un+1
−un
i
i
∆t
+
n
un
i+1 −ui−1
v 2∆x
−
n
n
n
2 ∆t ui+1 −2ui +ui−1
v 2
∆x2
= 0.
Beam-Warming
un+1
−un
i
i
∆t
+
n
n
3un
i −4ui−1 +ui−2
v
2∆x
−
n
n
n
2 ∆t ui −2ui−1 +ui−2
v 2
∆x2
= 0.
Crank-Nicholson
un+1
−un
i
i
∆t
+
n+1
un+1
i+1 −ui−1
θv 2∆x
+ (1 −
n
un
i+1 −ui−1
θ)v 2∆x
= 0.
home/lehre/VL-MHS-1-E/FOLIEN/VORLESUNG/9_UPWINDING/further_schemes.tex
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Upwinding - Universität Stuttgart

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