MagnetoHydroDynamics

1. MagnetoHydroDynamics Multigrid discretization and solution Achi Brandt The Weizmann Institute of Science

2. Resistive MHD Equations: Conservative Form

3. Time Dependent System • Implicit Each time step solve • Need solving to the level of incremental discretization error FMG for F cycle, FAS • Inside the MG solver: Defect corrections for higher order • Solver set-up – once per several (many?) time steps • Required: very high speed

4. Geometric MG, Finite differences Little geometric information Negligible set-up • Efficient discretization Simple stable high order Treatment of non ellipticity • Inexpensive coarse level Not Galerkin Polynomial interpolation • Fast DO loops on highly parallel, vector platforms • No linearization (FAS) Few, simple terms • Fast smoothing Principal-matrix DGS Possibly employing simple AMG cycle • Local refinement patches Option: piecewise characteristic alignment

5. Non-conservative MHD equations

6. MG Reduction Principles • Separate: interior boundary; shocks, … • Scale separation: Local processing at each scale • Principal terms at each scale

7. Principal Terms Principal terms on scale h: the larger coefficients in (or in replacing by , etc.) Important for: 1. Boundary conditions 2. Discretization (stability) 3. Relaxation forms 4. Interpolation

8. Principal-Term Relaxation fixed in calculating _____ Relax as FOR << 1 MAY CHANGE ON COARSE LEVELS ! Quasi-linear UU + … = f x Relax as fixed in calculating LIKE LINEAR ! AT ANY SCALE !

9. PDE Systems e.g., Stokes

10. on scale h h Principal terms of a system L are all terms which contribute to the principal part of det L on scale h h

11. Stokes

12. Potentially principal MHD matrix

13. Potentially principal MHD matrix _____ 0 _________ Principal, assuming

14. 0 Degeneracy! _____ 0 = O(h)? Principal, assuming

15. Distributive relaxation Stokes: Ghost variables Distribution matrix • Relax implicitly: distribute changes to (u, v, p) • Distributive GS: Smoothing rate like GS for

19. + O(h) ?

20. = Forward = Backward = Upstream

21. by Kacmarz relaxation

25. First order discretization = Forward = Backward = Upstream

26. Higher order discretization Higher order defect correction Efficient MG cycles

27. Higher order discretization • Converges to high order in a cycle or two Higher order defect correction Update once per cycle Efficient MG cycles • Does not converge to zero residuals • Accuracy checked by the FMG sequence

28. h 2h . . . h0/4 h0/2 h0 * * * * Full MultiGrid (FMG) algorithm

29. h 2h . . . h0/4 h0/2 h0 * * * * Full MultiGrid (FMG) algorithm

30. Line/Plane relaxation Double Variable U Semi-ellipticity Variable B piecewise • Semi coarsening AMG • Separate piecewise plane relaxation for each

31. Ghost variables

32. Ghost variables RELAX immediately distributing upon each move Distribution matrix

33. A relaxation sweep for 1. Relax starting with yielding 2. Relax starting with yielding

34. Double Variable U Semi-ellipticity Variable B • Separate piecewise plane relaxation for each e.g., by piecewise AMG cycles for • Super W cycles Characteristic-aligned discretization • Piecewise, in terms of each • Does not converge to zero residuals • Accuracy checked by the FMG sequence

35. Conservative equations NON-CONSERVATIVE DISTRIBUTIVE RELAXATION: Discontinuity management • Both and should be conservative and discontinuity-avoiding • Conservative fine-to-coarse residual restriction • Adaptive relaxation (near discontinuities)

36. MG Reduction Principles • Separate: interior boundary • Scale separation: Local processing at each scale • Principal terms at each scale • Design of discretizations, relaxation, interpolation, restriction in terms of the (scale-dependent) simple factors of the determinant of the principal quasi-linear operator scalar low-order factors, each featuring its own type (ellipticity measure, anisotropy,…) and characteristic directions • No linearizations • Design, debug - guided by quantitative predictions

37. THANK YOU!