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FLASH Workshop Hamburger Sternwarte , University of Hamburg, Feb 15 – Feb 16, 2012

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

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FLASH Workshop Hamburger Sternwarte , University of Hamburg, Feb 15 – Feb 16, 2012

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

  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 • Single-mode Rayleigh-Taylor Instability • Topfigures: • 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 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

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

  6. Part 1 • 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 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  7. Part 1 • 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: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

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

  9. Hydro_Unsplit Hydro Unit in FLASH 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 • 1st order Godunov, 2nd order MUSCL-Hancock, 3rd order PPM, 5th 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 Take a deep breath! 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. Linearized System • A primitive form: • where the coefficient matrix is FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  15. Corner Transport Upwind (CTU) Linear system in 3D FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  16. Corner Transport Upwind (CTU) Normal predictor Transverse corrector Linear system in 3D FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  17. Corner Transport Upwind (CTU) • Traditional approach (Colella 1990; Saltzman 1994) • Characteristic tracing for the normal predictor • Subsequent calls to Riemann solvers for transverse corrector Normal predictor Transverse corrector Linear system in 3D FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  18. Corner Transport Upwind (CTU) • 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! Normal predictor Transverse corrector Linear system in 3D FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  19. Linearized System, cont’d • 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:  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 Normal Predictor Characteristic Tracing 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 Characteristic Tracing for Normal Predictor Transverse Corrector FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  23. Full 3D CTU in USM Characteristic Tracing for Normal Predictor Transverse Corrector Full CTU diagonal coupling FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  24. Summary of Part 1 • 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 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  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 • CT scheme by Balsara and Spicer, 1998: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

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

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

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

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

  31. Summary of Part 2 • 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: • 3rd 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 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

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

  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 • 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 2nd order • 20% performance gain in using the full CTU scheme: • Various choices in runtime parameters FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

  40. Conclusion • 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 2nd 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 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

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

  42. New Upwind PPM for Slowly Moving Shock larger By Standard PPM with increasing By Standard PPM Upwind PPM 5th order WENO FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012

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

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

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