Time Discretization ( It’s about time…)

1. Time Discretization(It’sabout time…) Sauro Succi

2. Time Evolution Initial and Boundary Conditions

3. Time Marching: Formal

4. Time Marching: Matrix Exponential L banded b, L*L banded 2b+1, L*L*L banded 3b+2 …. UNPRACTICAL

5. Numerical Marching On the lattice x_j=j*d:

6. Numerical Marching On the lattice x_j=j*d:

7. Space-Time Connectivity

8. Numerical Marching Explicit Crank-Nicolson Fully Implicit

9. Numerical Marching Explicit Crank-Nicolson

10. Stability

11. Time marching: Explicit/Implicit methods Cranck-Nicolson Euler fwd - t+2dt - X ----------X----------X X ----------X----------X - t+dt - X ----------X----------X - t - X ----------X----------X X ----------X----------X Unconditionallystable --> large dt Non-local, matrix algebra, expensive Conditionally stable (|1+Ldt|<1) ---> small dt Matrix-free (local in spacetime) Simple and fast

12. Forward Euler Stability (for stable phys)

13. Crank-Nicolson Trapezoidal rule

14. Implicit=Non-local 3 spatialconnections 5spatialconnections 2p+1 spatialconnections

16. Implicit Diffusion

17. Matrix problem

18. Implicit Time-Marching

20. Matrix Algebra Algebraic Methods Direct Methods Iterative Methods

21. Matrix Inversion

22. Direct Methods 2. Backwardsweep: x 1. Forwardsweep: y

23. Gaussian elimination Zeroesincluded!

24. Multiple Dimensions

25. Memory is 1d: Addressing Sequentialindex:

26. Physical vs Logical indexing

27. Bandwidth minimization (d>1)

28. Iterative: Methods Splitting Relaxation Gradient ConjugateGradient No matrix inversion: only matrix*vector products

29. Iterative: Jacobi splitting Stoppingcriteria:

30. Gauss-Seidel: aggressive Jacobi Convergencecriteria:

31. Jacobi: does it converge? Diagonaldominant: Goodmatrixcondition:

32. Optimum condition=1 Badcondition>>1

33. Alternating Direction Implicit

34. Directional Splitting A sequence of Nyone-dimensional Problems of size NX:

35. Over/Under relaxation methods AcceleratedRelaxation Conservative, oscillatingconvergence Aggressive, montonicconvergence

36. Gradientmethods

37. Relaxation methods Fictitious time Iterationloop Convergencecriterion: Usuallyvery slow!

38. Steepest descent: gradient Residual=Gradient of the energy ris the gradient of E=1/2<x,Ax>-<b,x>

39. Steepest descent: “timestep”

40. Steepest Descent Convergentguaranteedonly for SPD matrices [(x,Ax)>0] Long-term slow becauser-->0 asAx-->b Round-off sensitive, needle-sensitive

41. Conjugate Gradient The steepestdescentisnot the shortestpath to the minimum g=r= Ax-b; <g,t>=0 c defined by <c,At> = 0 Oneshot on the local minimum ateachplanex_k,x_(k+1) t x x_k g c x_(k+1)

42. Conjugate Gradient The steepestdescentisnot the shortestpath to the minimum! 0. x0, p0=r0=A*x0-b The new searchdirection isbetweenr and oldp 1. a0=<r0,p0>/<r0,A*p0> x1=x0+a0*p0; r1=r0-a0*A*p0 Out? 3. b0=<r1*r1>/<r0*r0> p1 = r1+b0*p0 nextsearchdirection basedupon <p1,A*p0>=0 Go to 1 with p1 t x g c {p0,A*p0,A^2*p0 … A^N p0} is a Krylovsequence: Nstepconvergent for SPD

43. End of Lecture

44. Time Marching: Spectral Hermitian Stable/Unstable Example: diffusion

46. Operator Splitting (useful in d>1)

47. Non-commuting propagators

48. Trotter splitting