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42 Pages

### lecture16

Course: AE 232, Fall 2009
School: Caltech
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Word Count: 400

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Operator Today Splitting Motivation: Remove bottlenecks, stiffness / h^3,h^2 -&gt; 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 24 Ae232a. Tim Colonius 25 Ae232a. Tim Colonius 26 Ae232a. Tim Colonius 27 Ae232a. Tim Colonius 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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