New central and central discontinuous Galerkin schemes on overlapping cells of unstructured grids for solving ideal magnetohydrodynamic equations with globally divergence-free magnetic field

Abstract New schemes are developed on triangular grids for solving ideal magnetohydrodynamic equations while preserving globally divergence-free magnetic field. These schemes incorporate the constrained transport (CT) scheme of Evans and Hawley [34] with central schemes and central discontinuous Galerkin methods on overlapping cells which have no need for solving Riemann problems across cell edges where there are discontinuities of the numerical solution. These schemes are formally second-order accurate with major development on the reconstruction of globally divergence-free magnetic field on polygonal dual mesh. Moreover, the computational cost is reduced by solving the complete set of governing equations on the primal grid while only solving the magnetic induction equation on the polygonal dual mesh. Various numerical experiments are provided to validate the new schemes.

[1]  J. Stone,et al.  An unsplit Godunov method for ideal MHD via constrained transport , 2005, astro-ph/0501557.

[2]  Shengtai Li,et al.  High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method , 2008, J. Comput. Phys..

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

[4]  Michael Dumbser,et al.  Multidimensional Riemann problem with self-similar internal structure. Part II - Application to hyperbolic conservation laws on unstructured meshes , 2015, J. Comput. Phys..

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

[6]  J. Brackbill.  Fluid modeling of magnetized plasmas , 1985 .

[7]  Dinshaw S. Balsara.  Multidimensional HLLE Riemann solver: Application to Euler and magnetohydrodynamic flows , 2010, J. Comput. Phys..

[8]  M. Norman,et al.  ZEUS-2D : a radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. II : The magnetohydrodynamic algorithms and tests , 1992 .

[9]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

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

[11]  Zhiliang Xu,et al.  Divergence-Free WENO Reconstruction-Based Finite Volume Scheme for Solving Ideal MHD Equations on Triangular Meshes , 2011, 1110.0860.

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

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

[14]  Dinshaw Balsara,et al.  Second-Order-accurate Schemes for Magnetohydrodynamics with Divergence-free Reconstruction , 2003, astro-ph/0308249.

[15]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

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

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

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

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

[20]  Dinshaw S. Balsara,et al.  Multidimensional Riemann problem with self-similar internal structure. Part I - Application to hyperbolic conservation laws on structured meshes , 2014, J. Comput. Phys..

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

[22]  P. Londrillo,et al.  On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method , 2004 .

[23]  S. Orszag,et al.  Small-scale structure of two-dimensional magnetohydrodynamic turbulence , 1979, Journal of Fluid Mechanics.

[24]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

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

[26]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[27]  A. Harten,et al.  Multi-Dimensional ENO Schemes for General Geometries , 1991 .

[28]  Gérard Gallice,et al.  Roe Matrices for Ideal MHD and Systematic Construction of Roe Matrices for Systems of Conservation Laws , 1997 .

[29]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[30]  Chi-Wang Shu,et al.  Locally divergence-free discontinuous Galerkin methods for the Maxwell equations , 2004, Journal of Computational Physics.

[31]  S.,et al.  Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media , 1966 .

[32]  IXu-Dong Liu,et al.  Nonoscillatory High Order Accurate Self-similar Maximum Principle Satisfying Shock Capturing Schemes I , 1996 .

[33]  John Lyon,et al.  A simulation study of east-west IMF effects on the magnetosphere , 1981 .

[34]  Shengtai Li,et al.  A fourth-order divergence-free method for MHD flows , 2010, J. Comput. Phys..

[35]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[36]  K. Kusano,et al.  A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics , 2005 .

[37]  O. Friedrich,et al.  Weighted Essentially Non-Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids , 1998 .

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

[39]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[40]  Rémi Abgrall,et al.  Multidimensional HLLC Riemann solver for unstructured meshes - With application to Euler and MHD flows , 2014, J. Comput. Phys..

[41]  Yingjie Liu.  Central schemes on overlapping cells , 2005 .

[42]  C. Munz,et al.  Hyperbolic divergence cleaning for the MHD equations , 2002 .

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

[44]  Michael Dumbser,et al.  Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems , 2007, J. Comput. Phys..

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

[46]  Michael Dumbser,et al.  Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes - Speed comparisons with Runge-Kutta methods , 2013, J. Comput. Phys..

[47]  S. Osher,et al.  High-Resolution Nonoscillatory Central Schemes with Nonstaggered Grids for Hyperbolic Conservation Laws , 1998 .

[48]  Armin Iske,et al.  ADER schemes on adaptive triangular meshes for scalar conservation laws , 2005 .

[49]  Michael Dumbser,et al.  Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers , 2015, J. Comput. Phys..