Positivity-preserving DG and central DG methods for ideal MHD equations

Ideal MHD equations arise in many applications such as astrophysical plasmas and space physics, and they consist of a system of nonlinear hyperbolic conservation laws. The exact density @r and pressure p should be non-negative. Numerically, such positivity property is not always satisfied by approximated solutions. One can encounter this when simulating problems with low density, high Mach number, or much large magnetic energy compared with internal energy. When this occurs, numerical instability may develop and the simulation can break down. In this paper, we propose positivity-preserving discontinuous Galerkin and central discontinuous Galerkin methods for solving ideal MHD equations by following [X. Zhang, C.-W. Shu, Journal of Computational Physics 229 (2010) 8918-8934]. In one dimension, the positivity-preserving property is established for both methods under a reasonable assumption. The performance of the proposed methods, in terms of accuracy, stability and positivity-preserving property, is demonstrated through a set of one and two dimensional numerical experiments. The proposed methods formally can be of any order of accuracy.

[1]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[2]  J. M. Ball,et al.  SHOCK WAVES AND REACTION‐DIFFUSION EQUATIONS (Grundlehren der mathematischen Wissenschaften, 258) , 1984 .

[3]  Cheng Wang,et al.  Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations , 2012, J. Comput. Phys..

[4]  Chi-Wang Shu,et al.  On positivity preserving finite volume schemes for Euler equations , 1996 .

[5]  Chi-Wang Shu,et al.  Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations , 2005, J. Sci. Comput..

[6]  Dinshaw S. Balsara.  Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics , 2009, J. Comput. Phys..

[7]  Chi-Wang Shu,et al.  Central Discontinuous Galerkin Methods on Overlapping Cells with a Nonoscillatory Hierarchical Reconstruction , 2007, SIAM J. Numer. Anal..

[8]  Dinshaw S. Balsara,et al.  Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics , 2012, J. Comput. Phys..

[9]  Paul R. Woodward,et al.  A Simple Finite Difference Scheme for Multidimensional Magnetohydrodynamical Equations , 1998 .

[10]  G. Tóth.  The ∇·B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes , 2000 .

[11]  Xiangxiong Zhang,et al.  On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes , 2010, J. Comput. Phys..

[12]  Dinshaw S. Balsara,et al.  Maintaining Pressure Positivity in Magnetohydrodynamic Simulations , 1999 .

[13]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[14]  Kenneth G. Powell,et al.  AN APPROXIMATE RIEMANN SOLVER FOR MAGNETOHYDRODYNAMICS (That Works in More than One Dimension) , 1994 .

[15]  K. Waagan,et al.  A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics , 2009, J. Comput. Phys..

[16]  P. Janhunen,et al.  A Positive Conservative Method for Magnetohydrodynamics Based on HLL and Roe Methods , 2000 .

[17]  Katharine Gurski,et al.  An HLLC-Type Approximate Riemann Solver for Ideal Magnetohydrodynamics , 2001, SIAM J. Sci. Comput..

[18]  J. Brackbill,et al.  The Effect of Nonzero ∇ · B on the numerical solution of the magnetohydrodynamic equations☆ , 1980 .

[19]  Dinshaw S. Balsara,et al.  Notes on the Eigensystem of Magnetohydrodynamics , 1996, SIAM J. Appl. Math..

[20]  Dinshaw S. Balsara.  A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows , 2012, J. Comput. Phys..

[21]  J. Hawley,et al.  Simulation of magnetohydrodynamic flows: A Constrained transport method , 1988 .

[22]  Michael Dumbser,et al.  Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics , 2008, Journal of Computational Physics.

[23]  Sergey Yakovlev,et al.  Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field , 2011, J. Comput. Phys..

[24]  Xiangxiong Zhang,et al.  On maximum-principle-satisfying high order schemes for scalar conservation laws , 2010, J. Comput. Phys..

[25]  Liwei Xu,et al.  Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations , 2012, J. Comput. Phys..

[26]  Zhiliang Xu,et al.  Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells , 2009, J. Comput. Phys..

[27]  Sergey Yakovlev,et al.  Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations , 2013, J. Comput. Sci..

[28]  G. Tóth,et al.  Constraint in Shock-Capturing Magnetohydrodynamics Codes , 2007 .

[29]  D. Balsara,et al.  A Staggered Mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations , 1999 .