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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

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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

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

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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

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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)

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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

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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:

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

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

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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

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

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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

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

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

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

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

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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

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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!

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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

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

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

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

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

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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

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

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

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

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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:

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

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

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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

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

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

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

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

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

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

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

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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

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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

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

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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

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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

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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

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