Constraint preserving boundary conditions for the Ideal Newtonian MHD equations

Abstract We study and develop constraint preserving boundary conditions for the Newtonian magnetohydrodynamic equations and analyze the behavior of the numerical solution upon considering different possible options. We concentrate on both the standard ideal MHD system and the one augmented by a “pseudo potential” to control the divergence free constraint. We show how the boundary conditions developed significantly reduce the violations generated at the boundaries at the numerical level and how lessen their influence in the interior of the computational domain by making use of the available freedom in the equations.

[1]  L. Driel-Gesztelyi.  An Introduction to Magnetohydrodynamics , 2004 .

[2]  Relativistic magnetohydrodynamics in dynamical spacetimes: Numerical methods and tests , 2005, astro-ph/0503420.

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

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

[5]  Oscar Reula.  STRONGLY HYPERBOLIC SYSTEMS IN GENERAL RELATIVITY , 2004 .

[6]  Andreas Dedner,et al.  Transparent boundary conditions for MHD simulations in stratified atmospheres , 2001 .

[7]  H. Kreiss,et al.  Time-Dependent Problems and Difference Methods , 1996 .

[8]  S. Komissarov,et al.  Multi-dimensional Numerical Scheme for Resistive Relativistic MHD , 2007, 0708.0323.

[9]  E. Hirschmann,et al.  Relativistic MHD and black hole excision: Formulation and initial tests , 2005, gr-qc/0512147.

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

[11]  David Neilsen,et al.  Relativistic MHD with adaptive mesh refinement , 2006, gr-qc/0605102.

[12]  Department of Physics,et al.  WhiskyMHD: a new numerical code for general relativistic magnetohydrodynamics , 2007, gr-qc/0701109.

[13]  Axisymmetric simulations of magneto-rotational core collapse : dynamics and gravitational wave signal , 2005, astro-ph/0510184.

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

[15]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

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

[17]  Hui Li,et al.  CosmoMHD: A Cosmological Magnetohydrodynamics Code , 2006, astro-ph/0611863.

[18]  Numerical Simulations of MHD Turbulence in Accretion Disks , 2002, astro-ph/0203353.

[19]  P. Olsson.  Summation by parts, projections, and stability. II , 1995 .

[20]  J Korea,et al.  A Multidimensional Code for Isothermal Magnetohydrodynamic Flows in Astrophysics , 1998, astro-ph/9810356.

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

[22]  Dinshaw S. Balsara,et al.  Linearized Formulation of the Riemann Problem for Adiabatic and Isothermal Magnetohydrodynamics , 1998 .

[23]  Dinshaw Balsara,et al.  A Comparison between Divergence-Cleaning and Staggered-Mesh Formulations for Numerical Magnetohydrodynamics , 2003 .

[24]  Quentin F. Stout,et al.  An adaptive MHD method for global space weather simulations , 2000 .

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

[26]  Ian Hutchinson,et al.  Principles of Magnetohydrodynamics , 2005 .

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

[28]  C. Stivers.  Class , 2010 .

[29]  Dinshaw S. Balsara,et al.  Total Variation Diminishing Scheme for Relativistic Magnetohydrodynamics , 2001 .

[30]  P. Londrillo,et al.  An efficient shock-capturing central-type scheme for multidimensional relativistic flows. II. Magnetohydrodynamics , 2002 .

[31]  Y. Sekiguchi,et al.  Magnetohydrodynamics in full general relativity: Formulation and tests , 2005, astro-ph/0507383.

[32]  Timothy J. Barth,et al.  Numerical Methods for Gasdynamic Systems on Unstructured Meshes , 1997, Theory and Numerics for Conservation Laws.

[33]  S. Pennisi.  A covariant and extended model for relativistic magnetofluiddynamics , 1993 .

[34]  E. Hirschmann,et al.  Relativistic MHD and excision: formulation and initial tests , 2005, gr-qc/0512147.

[35]  Charles F. Gammie,et al.  HARM: A NUMERICAL SCHEME FOR GENERAL RELATIVISTIC MAGNETOHYDRODYNAMICS , 2003 .