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lecture16
Course: AE 232, Fall 2009School: Caltech
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Word Count: 400
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Operator Today Splitting Motivation: Remove bottlenecks, stiffness / h^3,h^2 -> h Each has same order of accuracy, but stability and efficiency of computation can be quite different 1 Ae232a. Tim Colonius Kinds of Splitting Temporal: Integrate different parts of RHS with different time marching methods Spatial: Factor matrices Other names: Fractional step method, operator splitting, factored...
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Operator Today
Splitting
Motivation: Remove bottlenecks, stiffness / h^3,h^2 -> h
Each has same order of accuracy, but stability and efficiency of computation can be quite different
1 Ae232a. Tim Colonius
Kinds of Splitting
Temporal: Integrate different parts of RHS with different time marching methods Spatial: Factor matrices Other names: Fractional step method, operator splitting, factored methods, etc.
2 Ae232a. Tim Colonius
Temporal Splitting
3 Ae232a. Tim Colonius
Temporal Splitting
4 Ae232a. Tim Colonius
Local Accuracy
5 Ae232a. Tim Colonius
Split multi-step method
6 Ae232a. Tim Colonius
A second-order split scheme
7 Ae232a. Tim Colonius
Analysis: stability
Look at circulant case (periodic BC). Neither necessary or sufficient for general case, but gives insight:
8 Ae232a. Tim Colonius
Explicit-Implicit Euler
9 Ae232a. Tim Colonius
||
2
R=2.0 R=2.5 R=1.5 R=1.0 R=0.5 R=0.1
C=1
kx
10 Ae232a. Tim Colonius
C
Stable
Ae232a. Tim Colonius
R
11
Explicit-Explicit Euler
R=0.1
R=0.5
R=1.0 R=0.5 R=2.0 R=2.5 R=50
C=1.0
kx
12
Ae232a. Tim Colonius
C
Stable
Ae232a. Tim Colonius
R
13
Trapezoidal-AB2
C
Trap/AB2 rough hand drawn curve see Lomax p 221 for exact curve
Euler/Euler
R
Ae232a. Tim Colonius
14
Spatial splitting: the diffusion equation in 2D
2 2 (I t(x + y ))un+1 = un jk 2 jk
2 (I tx ) (I ty ))un+1 = un jk jk w
Two tri-diagonal solves
15 Ae232a. Tim Colonius
16 Ae232a. Tim Colonius
u (x)
MxM
11 21 31 41 12 22 32 42 13 23 33 4317
Ae232a. Tim Colonius
18 Ae232a. Tim Colonius
u (y)
NxN
11 12 13 21 22 23 31 32 33 41 42 43
19
Ae232a. Tim Colonius
20 Ae232a. Tim Colonius
21 Ae232a. Tim Colonius
22 Ae232a. Tim Colonius
23 Ae232a. Tim Colonius
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25 Ae232a. Tim Colonius
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Conclusion:
28 Ae232a. Tim Colonius
A 2nd-order ADI Scheme
29 Ae232a. Tim Colonius
2nd-order ADI Scheme (contd)
30 Ae232a. Tim Colonius
Steady state solutions of factored systems
31 Ae232a. Tim Colonius
32 Ae232a. Tim Colonius
33 Ae232a. Tim Colonius
Stability of factored (ADI) schemes
34 Ae23...
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