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MA557/MA578/CS557 Lecture 32. Spring 2003 Prof. Tim Warburton [email protected]. Heat Equation. Recall Lecture 17 where we derived the advection-diffusion equation in 1D. The same type of derivation for the diffusion process of a concentrate C yields in 2D:. - PowerPoint PPT Presentation
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2
Heat Equation
• Recall Lecture 17 where we derived the advection-diffusion equation in 1D.
• The same type of derivation for the diffusion process of a concentrate C yields in 2D:
2
2 22
2 2
where:
diffusion constant
CD C
t
x y
D
3
Discretization of the Laplacian
• The LDG, semidiscrete, equation satisfies:
• Zero Dirichlet boundary conditions (concentration pinned to zero at boundary):
• Zero Neumann boundary conditions (insulated boundaries):
N D N DDt
x x y y
CD D +D D C
D N D NDt
x x y y
CD D +D D C
4
Heat Equation
• Our first approach will be an explicit time integration of the equation (using zero Neumann bc):
n
n
1
=
for 1:
=
end
D N D Nx x y y
n
rk RKorder
dtD
rk
σ C
σ C D D D D σ
C σ
5
Explicit Numerical Heat Equation
• The trouble is that the discrete diffusion matrix has a large spectral radius:
• We do know that (using the integration by parts formula for the DG operator) that the operator is non-positive.
• We also need to fit the eigenspectrum in the appropriate stability region:
D N D ND x x y yA D D +D D
6
2
2
2
2 2
2 2
2 2
2 22 2
2
2
2 2
2 2
22
222 2
42
2
, ,
using po
N Nx xN NLy y
L
NxNy
L
N Nx yL L
N Nx yL L
y
L L
x y
L LL L
L
D
D
D D
D D
D Dx y
kDx y
Dpkh
x
D DC AC C C
D D
DC
D
D C D C
D C D C
C CF C F C
C CF C F C
C lynomial inverse inequalities
7
22
2
2
42
2
4
2
,
,max
,
LL
L
L
Dpkh
Dpkh
C
C AC C
C ACA
C C
We also know that all the eigenvalues are non-positive real numbers and dt*rho(A) must fit in the stability region..
2
4 which can be small
hdt
Dp
8
Comment
• We have shown that using an explicit time integration scheme may require very small time steps…
• The root cause of this is that information diffuses through the data domain very quickly..
• We will investigate some schemes which solve globally coupled systems.
9
Backwards Euler Time Integration• Suppose we consider the backwards Euler scheme:
• Iterator:
1
1n n
D N D N nx x y yD
dt
C CM M D D D D C
1
1
1
where:
and:
thus by non-negativity of
(for theory, consider this in the PKDO basis =I)
1 , 0 1
n n
D N D Nx x y y
dt
D
dt D
C AC
A M L M
L M D D D D
L
M
A A
10
Backwards Euler Time Integration
• Suppose we consider the backwards Euler scheme:
• Thus the only conditions for stability are:
• However, accuracy depends on dt. The discretization of the time derivative has a first order truncation error:
1
1n n
D N D N nx x y yD
dt
C CM M D D D D C
, 0dt D
1 1n n n
O dtdt t
C C C
11
Crank-Nicholson
• We replace the right hand side of the discrete scheme with:
1 1
1
1
2
where: 2 2
and:
n n n nD N D Nx x y y
n n
D N D Nx x y y
Ddt
dt dt
D
C C C CM M D D D D
C AC
L LA M M
L M D D D D
12
Diagonalize
• If we apply a change of basis so that the inv(M)*L is diagonal then we decouple the iterator into a sequence of independent scalar iterators:
1
1
where:
1 12 2
n nm m m
m mm
C C
dt dt
13
Quick Result• We can bound the iterator multiplier:
• As long as both the eigenvalues and dt are non-negative then the multiplier is bounded by one and the scheme is stable.
• We have previously shown that L is a non-negative operator.
2
0
1
12
01
max 1
f
df
d
f
14
Crank-Nicholson Accuracy
• We can briefly compute the temporal accuracy:
3
1
2 23
2
11
2
2 2
2 2 4 2
Temporal truncation error
n n
n n
n
n
dt dt
d
O dt
O dt
tdt O dt dt
dt
1 1
1
C AC
L LC M M C
M L M L M L1 1 C
1 M L C
15
ESDIRK4
• A new implicit RK scheme with a lower triangle Butcher tableau has been proposed by Carpenter et al. http://fun3d.larc.nasa.gov/papers/carpenter.pdf
• The linear system to invert at each sub stage is the same.
• The RK scheme comes with an embedded scheme to provide s’th order and (s-1)’th order time approximations:
16
4th Order ESDIRK4 Implicit RK Schemeinitializer + 5 stages
• Iterator: (Ctilde are intermediate values of C, Chat6 is the embedded 3rd order approximation of Cn+1)
1
2 1 221
3 1 2 331 32
4 1 2 3 441 42 43
5 1 2 3 4 551 52 53 54
6 1 2 3 4 561 62 63 64 65
n
n
n
n
n
n
dt a
dt a a
dt a a a
dt a a a a
dt a a a a a
C C
MC MC LC LC
MC MC LC LC LC
MC MC LC LC LC LC
MC MC LC LC LC LC LC
MC MC LC LC LC LC LC
6 1 2 3 4 5 6
1 2 3
1 6
5 6
6
4ˆ ˆ ˆ ˆ ˆ ˆˆ
n
n dt b b b b b b
C C LC LC LC LC LC L
LC
M
C C
CM
17
4th Order ESDIRK4 Implicit RK Scheme
• Iterator: (Ctilde are intermediate values of C, Chat5 is the embedded 3rd order approximation of Cn+1)
1
2 121
3 1 231 32
4 1 2 341 42 43
5 1 2 3 451 52 53 54
6 1 2 3 4 51 2 3 4
6
5
ˆ
n
n
n
n
n
n
dt dt a
dt dt a a
dt dt a a a
dt dt a a a a
dt dt b b b b b
C C
M L C MC LC
M L C MC LC LC
M L C MC LC LC LC
M L C MC LC LC LC LC
M L C MC LC LC LC LC LC
MC
1 2 3 4 5 61 2 3 4 6
6
5
1
ˆ ˆ ˆ ˆ ˆ ˆ
n
n dt b b b b b b
C LC LC LC LC LC LM C
C C
18
4th Order ESDIRK4 Implicit RK Scheme
• If L is a linear operator then:
1
2 121
3 1 231 32
4 1 2 341 42 43
5 1 2 3 451 52 53 54
6 1 2 3 4 51 3
6
2 4 5
ˆˆ n
n
n
n
n
n
n
dt dt a
dt dt a a
dt dt a a a
dt dt a a a a
dt d
dt b
t b b b b b
C C
M L C MC L C
M L C MC L C C
M L C MC L C C C
M L C MC L C C C C
M L C MC L C C C C C
M MC C L
1 2 3 4
1 6
5 61 2 3 4 5 6
ˆ ˆ ˆ ˆ ˆ
n
b b b b b
C C C C
C
C C
C
19
Coefficients
Gamma = 1/4
a21 = 1/4
a31 = 8611/62500
a32 = -1743/31250
a41 = 5012029/34652500
a42 = -654441/2922500
a43 = 174375/388108
a51 = 15267082809/155376265600
a52 = -71443401/120774400
a53 = 730878875/902184768
a54 = 2285395/8070912
bhat1 =4586570599/29645900160bhat2 = 0bhat3 = 178811875/945068544 bhat4 = 814220225/1159782912 bhat5 = -3700637/11593932bhat6 = 61727/225920
b1 = 82889/524892
b2 = 0
b3 = 15625/83664
b4 = 69875/102672
b5 = -2260/8211
20
Implementation Details
• As they are computed we are required to store Ctilde1, Ctilde2, …, Ctilde6 until the end of the 6th ESDIRK stage.
• Each stage 2,3,4,5,6 requires the solution of the same linear system, with different data on the right hand side.
• Chat6 may be computed without solving a linear system.
• The systems can be solved using an iterative (say conjugate gradient) method. i.e. the L matrix does not need to be computed and stored.
• We need to be able to compute (M+dt*gamma*L)*x for a given vector x.
• ||Ctilde6-Chat6|| will give an estimate of the difference between a 3rd order and the 4th order approximation in time. This can be used to estimate the error made in this time step and can suggest a change in dt.
21
Solving the System
• We are going to use an iterative method to solve the following sequence of symmetric matrix problems:
6 1 2 3 4 5 6
1 2 3
1
2 121
3 1 231 32
6 1 2 3 4 51 2 3 4 5
6
1
4 5
6
ˆ ˆ ˆ ˆ ˆ ˆˆ
n
n
n
D N D Nx x y
n
y
n
n
dt dt a
dt dt a a
dt d
dt b b b b
t b b b b
b b
b
L M
C C
M L C MC L C
M L C MC L C C
M L C MC L C C C C C
M MC
D D
C L C C C C C
D
C C
C
D
22
Summary of Temporal Implicit Schemes
• Backwards Euler is unconditionally stable for non-negative diffusion parameter D (i.e. any dt>=0) and first order in dt.
• Crank-Nicholson is unconditionally stable for non-negative diffusion parameter D (i.e. any dt>=0) and second order in dt.
• ESDIRK4 – generalizes to fourth order in dt.
11
2 2n ndt dt
L L
C M M C
11n ndt C M L MC
D N D Nx x y yD L M D D D D
23
Homework 10
Q1) Modify/use the umDIFFUSION.zip files to solve the diffusion equation with the following time integrators:
a) Backwards Eulerb) Crank-Nicholsonc) ESDIRK4
Q2) Create a domain and determine convergence rates in h for p=1,3,5,7 for a small dt (i.e. make sure the convergence does not bottom out above 1e-10). Repeat for two of the above time integrators.
Q3) Choose p=7 and h small and verify rates of convergence in dt for integration to some fixed time T. Use the one time integrator you did not use in Q2.
24
Homework 10
• This homework can be completed in pairs or individually.
• This homework is due Monday 04/21/03
• Remember – no more than 5 email questions will be replied to for this homework.
25
Driver for Backwards Euler Diffusion Scheme(umDIFFUSIONdemo.m)
26
Time Loop (umDIFFUSIONrun.m)
1-12) Set parameters
14-16) Set initial conditions
19) Set solver parameters
22-40) Time stepping loop
28) Solves linear system using conjugate gradient
32-40) Plot solution and compute integral of C
27
umDIFFUSIONop.m
• Set up storage for computing: (M*C +dt*gamma*D* L*C)
28
umDIFFUSIONop.m
15-25) compute qx,qy using Neumann bcs
27-40) compute div(qx,qy) using Dirichlet bcs
42-48) compute (M*C+gamma*dt*D*L*C)
29
Demo Run
30
Discussion of Final Project
• Now is the time to ready your final projects. • Submit a one page description of the PDE you intend to
discretize with DG by 04/21/03
• This is a proposal and may be rejected – requiring a resubmission or assignment of a set of PDEs.
• Include:– List of PDEs to be discretized
– List boundary conditions and initial conditions to be discretized
– Relevance to your own research (if any)
– List of group members (max 3, with one proposal per group)
– Interesting issues (application of PML, creation of specific PML, specific physical application/model…)