On the origin of divergence errors in MHD simulations and consequences for numerical schemes

This paper investigates the origin of divergence errors in MHD simulations. For that purpose, we introduce the concept of discrete involutions for discretized conservation laws. This is done in analogue to the concept of involutions for hyperbolic conservation laws, introduced by Dafermos. By exploring the connection between discrete involutions and resonance, especially for constrained transport like MHD, we identify the lack of positive central viscosity and the assumption of one-dimensional physics in the calculation of intercell fluxes as the main sources of divergence errors. As an example of the consequences for numerical schemes, we give a hint how to modify Roe-type schemes in order to decrease the divergence errors considerably and, thus, stabilize the scheme.

[1]  F. Kemm.  A comparative study of TVD‐limiters—well‐known limiters and an introduction of new ones , 2011 .

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

[3]  B. M. Marder,et al.  A method for incorporating Gauss' lasw into electromagnetic pic codes , 1987 .

[4]  Claus-Dieter Munz,et al.  A Finite-Volume Method for the Maxwell Equations in the Time Domain , 2000, SIAM J. Sci. Comput..

[5]  Phillip Colella,et al.  A Higher-Order Godunov Method for Multidimensional Ideal Magnetohydrodynamics , 1994, SIAM J. Sci. Comput..

[7]  Eitan Tadmor,et al.  Nonoscillatory Central Schemes for One- and Two-Dimensional Magnetohydrodynamics Equations. II: High-Order SemiDiscrete Schemes , 2006, SIAM J. Sci. Comput..

[8]  Claus-Dieter Munz,et al.  Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model , 2000 .

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

[10]  C. Angelopoulos.  High resolution schemes for hyperbolic conservation laws , 1992 .

[11]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

[12]  Constantine M. Dafermos.  Quasilinear Hyperbolic Systems with Involutions , 1986 .

[13]  E. Tadmor,et al.  Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations , 2012 .

[14]  James A. Rossmanith,et al.  An Unstaggered, High-Resolution Constrained Transport Method for Magnetohydrodynamic Flows , 2006, SIAM J. Sci. Comput..

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

[16]  Dinshaw Balsara,et al.  Divergence-free adaptive mesh refinement for Magnetohydrodynamics , 2001 .

[17]  Hans d,et al.  Multi-dimensional upwind constrained transport on unstructured grids for 'shallow water' magnetohydrodynamics , 2001 .

[18]  D. Kröner,et al.  Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system , 2005 .

[19]  N. Risebro,et al.  STABLE UPWIND SCHEMES FOR THE MAGNETIC INDUCTION EQUATION , 2009 .

[20]  R. I. Klein,et al.  An unsplit, cell-centered Godunov method for ideal MHD , 2005 .

[21]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

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

[23]  Philip L. Roe,et al.  An upwind scheme for magnetohydrodynamics , 1995 .

[24]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[25]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[26]  Friedemann Kemm,et al.  A Carbuncle Free Roe-Type Solver for the Euler Equations , 2008 .

[27]  S. Mishra,et al.  Splitting based finite volume schemes for ideal MHD equations , 2009, J. Comput. Phys..

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

[29]  Claus-Dieter Munz,et al.  Maxwell's equations when the charge conservation is not satisfied , 1999 .

[30]  Bernd Einfeld.  On Godunov-type methods for gas dynamics , 1988 .

[31]  Eitan Tadmor,et al.  Hyperbolic Problems: Theory, Numerics and Applications , 2009 .

[32]  Manuel Torrilhon.  Zur Numerik der idealen Magnetohydrodynamik , 2004 .

[33]  P. Roe,et al.  A Solution-Adaptive Upwind Scheme for Ideal Magnetohydrodynamics , 1999 .

[34]  Manuel Torrilhon,et al.  Locally Divergence-preserving Upwind Finite Volume Schemes for Magnetohydrodynamic Equations , 2005, SIAM J. Sci. Comput..

[35]  Eitan Tadmor,et al.  Constraint Preserving Schemes Using Potential-Based Fluxes. II. Genuinely Multidimensional Systems of Conservation Laws , 2011, SIAM J. Numer. Anal..

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

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

[38]  Peter A. Gilman,et al.  Magnetohydrodynamic “Shallow Water” Equations for the Solar Tachocline , 2000 .

[39]  Manuel Torrilhon,et al.  Constraint-Preserving Upwind Methods for Multidimensional Advection Equations , 2004, SIAM J. Numer. Anal..

[40]  M. Breuß.  The correct use of the Lax–Friedrichs method , 2004 .