Report

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 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 Part 1 Dimensionally Split vs. Unsplit??? FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 1 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 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 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 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 Part 2 Unsplit Hydro/MHD Solvers & Algorithms FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Hydro Unit in FLASH Hydro_Unsplit FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 Unsplit Formulations FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 MHD Governing Equations Conservative variables and fluxes: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Linearized System A primitive form: where the coefficient matrix is FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Corner Transport Upwind (CTU) FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Corner Transport Upwind (CTU) Normal predictor Transverse corrector FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 Single-step data Reconstruction-evolution in USM FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Characteristic tracing for Transverse corrector A jump relationship: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Reduced 3D CTU in USM FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Full 3D CTU in USM FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 Part 2 Divergence-Free fields: Constrained Transport (CT) MHD FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 2 CT scheme by Balsara and Spicer, 1998: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 2: recall… Conservative variables and fluxes: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 Part 2 Small angle advection of the 2D field loop: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 2 Small angle advection of the 3D field loop: FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 Part 3 Verification, convergence, and performance FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Part 3 FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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: CPU F-ctu » 0.8 CPU R-ctu Various choices in runtime parameters FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 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 Thank You Questions? FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 New Upwind PPM for Slowly Moving Shock larger By Standard PPM Upwind PPM Standard PPM with increasing By 5th order WENO FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 New Upwind PPM for Slowly Moving Shock larger By Standard PPM Standard PPM with increasing By Lee, 2010, 5th Astronum Proceeding; Lee, 2011, in preparation Upwind PPM 5th order WENO FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Block and Mesh Packages q q q 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 AMR with variable patch PARAMESH size - CHOMBO FLASH Workshop Hamburger Sternwarte, University of Hamburg, Feb 15 – Feb 16, 2012 Uniform Grid