1. FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 A Solution Accurate, Efficient and Stable Unsplit Staggered Mesh MHD Solver in FLASH Dongwook Lee University of Chicago The Flash Center for Computational Science

2. Outline  Split vs. unsplit formulations  Unsplit solvers in FLASH (UHD & USM)  CFL stability (reduced or full?)  Reduced/Full corner-transport-upwind (CTU) for 3D  Divergence-free magnetic fields for USM-MHD  constrained-transport (CT)  Verifications, convergence, performance  Runtime parameters  Summary FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

3. Part 1 Dimensionally Split vs. Unsplit??? FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

4. Part 1 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  Single-mode Rayleigh-Taylor Instability  Top figures:  Dimensionally split using PLM, PPM+old limiter, PPM+new limiter  high-wavenumber instabilities grow  Bottom figures:  Dimensionally unsplit using PLM, PPM+old limiter, PPM+new limiter  high-wavenumber instabilities suppressed  the split solvers experience high compressions and expansions in subsequent directional sweeps where there is a local high strain rate  Almgren et al, ApJ, 715, 2010  Single-mode Rayleigh-Taylor Instability  Top figures:  Dimensionally split using PLM, PPM+old limiter, PPM+new limiter  high-wavenumber instabilities grow  Bottom figures:  Dimensionally unsplit using PLM, PPM+old limiter, PPM+new limiter  high-wavenumber instabilities suppressed  the split solvers experience high compressions and expansions in subsequent directional sweeps where there is a local high strain rate  Almgren et al, ApJ, 715, 2010

5. Part 1 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  Weakly magnetized 2D field loop  Gardiner and Stone 2005 (JCP); Lee and Deane 2009 (JCP)  Weakly magnetized 2D field loop  Gardiner and Stone 2005 (JCP); Lee and Deane 2009 (JCP)

6. Part 1 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  8-wave split MHD scheme (Powell et al. 1999) at t=2.0  Unsplit staggered mesh MHD scheme (Lee and Deane, 2009) at t=2.0

7. Part 1 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  What is wrong with the split formulation for MHD?  In the split formulation, you cannot correctly include terms proportional to  Gardiner and Stone (2005)  Dynamics of in-plane magnetic fields in x and y directions are ruined from erroneous growth of magnetic field in z direction:  What is wrong with the split formulation for MHD?  In the split formulation, you cannot correctly include terms proportional to  Gardiner and Stone (2005)  Dynamics of in-plane magnetic fields in x and y directions are ruined from erroneous growth of magnetic field in z direction:

8. Part 2 Unsplit Hydro/MHD Solvers & Algorithms FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

9. Hydro Unit in FLASH Hydro_Unsplit FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

10. Unsplit Staggered Mesh (USM) MHD Solver  Shock-capturing high-order Godunov Riemann solver (Lee & Deane, JCP, 2009; Lee 2012, to be submitted)  Finite volume method  New data reconstruction-evolution algorithm for high-order accuracy  Adaptive mesh refinement, uniform grid  1 st order Godunov, 2 nd order MUSCL-Hancock, 3 rd order PPM, 5 th Order WENO  Approximate Riemann solvers: Roe, HLL, HLLC, HLLD, Marquina, modified Marquina, Local Lax-Friedrichs  Monotonicity preserving upwind PPM slope limiter for MHD (Lee, 2010, Astronum)  Divergence of magnetic fields is numerically controlled on a staggered grid, using a constrained transport (CT) method (Evans & Hawley, 1998)  Wide ranges of plasma flows  Full Courant stability limit (CFL ~ 1 for 3D) FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

11. Unsplit Formulations FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

12. MHD Governing Equations  MHD system of equations:  This can be written in a simple matrix form: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

13. MHD Governing Equations  Conservative variables and fluxes: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

14.  A primitive form: where the coefficient matrix is Linearized System FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

15. Corner Transport Upwind (CTU) FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

16. Corner Transport Upwind (CTU) Normal predictor Transverse corrector FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

17. Corner Transport Upwind (CTU) Normal predictor Transverse corrector FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  Traditional approach (Colella 1990; Saltzman 1994)  Characteristic tracing for the normal predictor  Subsequent calls to Riemann solvers for transverse corrector  Traditional approach (Colella 1990; Saltzman 1994)  Characteristic tracing for the normal predictor  Subsequent calls to Riemann solvers for transverse corrector

18. Corner Transport Upwind (CTU) Normal predictor Transverse corrector FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  Traditional approach (Colella 1990; Saltzman 1994)  Characteristic tracing for the normal predictor  Subsequent calls to Riemann solvers for transverse corrector  Traditional approach (Colella 1990; Saltzman 1994)  Characteristic tracing for the normal predictor  Subsequent calls to Riemann solvers for transverse corrector  New approach (Lee and Deane 2009):  Characteristic tracing for BOTH normal predictor and transverse corrector!  New approach (Lee and Deane 2009):  Characteristic tracing for BOTH normal predictor and transverse corrector!

19.  A primitive form: where the coefficient matrix is  First consider the evolution in the x-normal direction and treat the normal magnetic field separately from the other variables: Linearized System, cont’d  Normal predictor  MHD source term FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

20. Single-step data Reconstruction-evolution in USM FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

21. Characteristic tracing for Transverse corrector  A jump relationship: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

22. Reduced 3D CTU in USM FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

23. Full 3D CTU in USM FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

24. Summary of Part 1 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  New approach of using characteristic tracing for BOTH normal predictor and transverse corrector  Reduced 3D CTU  A direct extension of 2D CTU to 3D  Requires 3 Riemann solves for 3D (6-ctu needs 6 Riemann solves)  Only including second cross derivatives  CFL limit ~ 0.5  Full 3D CTU  Full considerations of accounting for third cross derivatives  Requires 3 Riemann solves for 3D (12-ctu needs 12 Riemann solves)  CFL limit ~ 1.0  20% relative performance gain compared to reduced 3D CTU

25. Part 2 Divergence-Free fields: Constrained Transport (CT) MHD FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

26. Part 2 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  CT scheme by Balsara and Spicer, 1998:

27. Part 2: recall…  Conservative variables and fluxes: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

28. Part 2 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  New upwind biased modified electric field construction(upwind-MEC), Lee 2012:

29. Part 2 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  Small angle advection of the 2D field loop:

30. Part 2 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  Small angle advection of the 3D field loop:

31. Summary of Part 2 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  Three CT schemes were discussed:  Standard CT scheme by Balsara and Spicer, 1998:  Takes a simple arithmetic averaging  Lacks numerical diffusion for magnetic fields advection  Modified electric field construction (MEC) scheme by Lee and Deane, 2009:  3 rd order accurate in space  Not enough numerical diffusion for field advection  Upwind biased MEC (upwind-MEC) scheme by Lee, 2012 (to be submitted)  Upwind scheme of MEC  Added numerical diffusion to stabilize field advection

32. Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Verification, convergence, and performance

33. Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

34. Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

35. Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

36. Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

37. Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

38. Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

39. Summary of Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  Verification tests for the reduced/full 3D CTU schemes:  CFL=0.95 for all 3D simulations using the full CTU scheme  CFL=0.475 for the reduced CTU scheme  They both converge in 2 nd order  20% performance gain in using the full CTU scheme:  Various choices in runtime parameters

40. Conclusion FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012  Directionally split vs. unsplit formulations for hydro and MHD  Unsplit hydro/MHD solvers in FLASH4 (also FLASH3 in part)  The reduced and full 3D CTU algorithms  Upwind-MEC scheme for MHD  Stable solutions with 2 nd order convergence with CFL=0.95  20% performance gain in the full CTU scheme over the reduced CTU scheme  Work in progress:  Fully implicit Jacobian-Free Newton-Krylov implicit solver for the unsplit solvers  More HEDP capabilities for the USM solver

41. Thank You FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Questions?

42. New Upwind PPM for Slowly Moving Shock Upwind PPM5 th order WENO Standard PPM Standard PPM with increasing By larger By FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

43. New Upwind PPM for Slowly Moving Shock Upwind PPM5 th order WENO Standard PPM Standard PPM with increasing By Lee, 2010, 5 th Astronum Proceeding; Lee, 2011, in preparation larger By FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

44. Block and Mesh Packages Uniform Grid AMR with variable patch size - CHOMBO qMesh package can be selected at configuration time qThe basic abstraction is a block of interior cells surrounded by guard cells qGrid unit makes sure that blocks are self contained before being given to the solvers Oct tree based AMR - PARAMESH FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012