1. Computational Fluid Dynamics I PIW Numerical Methods for Parabolic Equations Instructor: Hong G. Im University of Michigan Fall 2005

2. Computational Fluid Dynamics I PIW One-Dimensional Problems Explicit, implicit, Crank-Nicolson Accuracy, stability Various schemes Keller Box method and block tridiagonal system Multi-Dimensional Problems Alternating Direction Implicit (ADI) Approximate Factorization of Crank-Nicolson Outline Solution Methods for Parabolic Equations

3. Computational Fluid Dynamics I PIW Numerical Methods for One-Dimensional Heat Equations

4. Computational Fluid Dynamics I PIW which is a parabolic equation requiring In this lecture, we consider a model equation Initial Condition Boundary Condition (Dirichlet) Boundary Condition (Neumann) or and

5. Computational Fluid Dynamics I PIW Explicit: FTCS j  1 j j+1 n n+1 Explicit Method: FTCS - 1

6. Computational Fluid Dynamics I PIW Modified Equation where - Accuracy - If then - No odd derivatives; dissipative Explicit Method: FTCS - 2

7. Computational Fluid Dynamics I PIW Stability: von Neumann Analysis (Recall Lecture 2, p. 60) Fourier Condition Explicit Method: FTCS - 3

8. Computational Fluid Dynamics I PIW

9. PIW

10. PIW Domain of Dependence for Explicit Scheme BC x t Initial Data h P tt Boundary effect is not felt at P for many time steps This may result in unphysical solution behavior Explicit Method: FTCS - 4

11. Computational Fluid Dynamics I PIW Implicit Method: Laasonen (1949) j  1 j j+1 n n+1 Tri-diagonal matrix system Implicit Method - 1

12. Computational Fluid Dynamics I PIW - The + sign suggests that implicit method may be less accurate than a carefully implemented explicit method. Modified Equation Implicit Method - 2 Amplification Factor (von Neumann analysis) Unconditionally stable

13. Computational Fluid Dynamics I PIW Crank-Nicolson Method (1947) j  1 j j+1 n n+1 Tri-diagonal matrix system Crank-Nicolson - 1

14. Computational Fluid Dynamics I PIW - Second-order accuracy Modified Equation Crank-Nicolson - 2 Amplification Factor (von Neumann analysis) Unconditionally stable

15. Computational Fluid Dynamics I PIW Generalization j  1 j j+1 n n+1 Combined Method A - 1

16. Computational Fluid Dynamics I PIW Other Special Cases (for further accuracy improvement): Combined Method A - 2 (a) If (b) If and Modified Equation

17. Computational Fluid Dynamics I PIW  unconditionally stable Combined Method A - 3 Stability Property  stable only if

18. Computational Fluid Dynamics I PIW Generalized Three-Time-Level Implicit Scheme: Richtmyer and Morton (1967) j  1 j j+1 n n+1 Combined Method B - 1 n1n1

19. Computational Fluid Dynamics I PIW Modified Equation: Combined Method B - 2 Special Cases: (a) If (b) If

20. Computational Fluid Dynamics I PIW Richardson Method: A Case of Failure Richardson Method Similar to Leapfrog but unconditionally unstable! j  1 j j+1 n n+1 n1n1

21. Computational Fluid Dynamics I PIW The Richardson method can be made stable by splitting by time average j  1 j j+1 n n+1 DuFort-Frankel - 1 n1n1

22. Computational Fluid Dynamics I PIW Modified Equation (Recall Homework #1) DuFort-Frankel - 2 Amplification factor Unconditionally stable Conditionally consistent

23. Computational Fluid Dynamics I PIW FTCS Stable for BTCS Unconditionally Stable Crank-Nicolson Unconditionally Stable Richardson Unconditionally Unstable Parabolic Equation - Summary

24. Computational Fluid Dynamics I PIW DuFort-Frankel Unconditionally Stable, Conditionally Consistent 3-Level Implicit Unconditionally Stable Parabolic Equation - Summary

25. Computational Fluid Dynamics I PIW The Keller Box Method (1970) Keller Box - 1 - Implicit with Basic Concept: Define yielding

26. Computational Fluid Dynamics I PIW In discretized form Keller Box - 2 j  1 j n+1 n  t n+1 hjhj j  1 j n+1 n where

27. Computational Fluid Dynamics I PIW Substituting Keller Box - 3 or

28. Computational Fluid Dynamics I PIW Adding boundary conditions, e.g. Keller Box - 4

29. Computational Fluid Dynamics I PIW In Matrix Form Keller Box - 5 D1D1 A1A1 A2A2 D2D2 B2B2 B3B3 D3D3 A3A3 DJDJ BJBJ Block Tridiagonal Matrix  Appendix, Linpack

30. Computational Fluid Dynamics I PIW Modified Keller Box Method – Express in terms of Keller Box - 6 Starting with original Keller box method: Eliminating from (b) using (a) (a) (b)

31. Computational Fluid Dynamics I PIW Further elimination of yields (Tannehill, p. 136) Keller Box - 7 Tridiagonal Matrix  Thomas Algorithm

32. Computational Fluid Dynamics I PIW Notes on Keller Box Method Keller Box - 8 1.Second-order accurate in time and space 2.Accuracy is preserved for nonuniform grids 3.More operations per timestep compared to Crank-Nicolson

33. Computational Fluid Dynamics I PIW Numerical Methods for Multi-Dimensional Heat Equations

34. Computational Fluid Dynamics I PIW Applying forward Euler scheme: Consider a 2-D heat equation If Explicit Method - 1

35. Computational Fluid Dynamics I PIW In matrix form Explicit Method - 2

36. Computational Fluid Dynamics I PIW Explicit Method - 3 Von Neumann Analysis Worst case

37. Computational Fluid Dynamics I PIW Explicit method for multi-dimensional heat equation is not desirable. Explicit Method - 4 Stability Condition for Heat Equation (2-D) (3-D) (1-D)

38. Computational Fluid Dynamics I PIW Too expensive!! Crank-Nicolson Crank-Nicolson Method for 2-D Heat Equation If

39. Computational Fluid Dynamics I PIW Fractional Step: ADI - 1 A Clever Remedy – Alternating Direction Implicit (ADI) Step 1: Step 2: Combining the two becomes equivalent to:

40. Computational Fluid Dynamics I PIW ADI - 2 Computational Molecules for ADI Method n n+1/2 n+1 i i+1 i1i1 j+1 j1j1 j

41. Computational Fluid Dynamics I PIW Stability Analysis: ADI - 3 ADI Method is accurate Similarly,

42. Computational Fluid Dynamics I PIW ADI - 4 Combining Unconditionally stable! Unfortunately, a 3-D version does not have the same desirable stability properties.  Conditionally stable and

43. Computational Fluid Dynamics I PIW ADI - 5 Advantages: 1.Stability limits of 1-D case apply. 2.Different  t can be used in x and y directions. 1. Cannot be directly extended to 3-D problems.  Approximate factorization Limitations:

44. Computational Fluid Dynamics I PIW Approximate Factorization - 1 The Crank-Nicolson for heat equation becomes Define which can be rewritten as

45. Computational Fluid Dynamics I PIW Approximate Factorization - 2 Factoring each side or

46. Computational Fluid Dynamics I PIW Approximate Factorization - 3 Final factored form of the discrete equation at an accuracy of Note that the ADI method can be written as which is equivalent to factorized Crank-Nicolson

47. Computational Fluid Dynamics I PIW Approximate Factorization - 4 Two-step algorithm can be written into two steps: each of which can be solved by TDMA (Thomas algorithm). (Step 2) (Step 1)

48. Computational Fluid Dynamics I PIW Approximate Factorization - 5 Note: Boundary condition for is needed at This can be determined from For example, at Similarly, at

49. Computational Fluid Dynamics I PIW Approximate Factorization - 6 Generalized 3-D Algorithm: unconditionally stable, where (Step 2) (Step 1) (Step 3)