Divergence-free adaptive mesh refinement for Magnetohydrodynamics

Abstract Several physical systems, such as nonrelativistic and relativistic magnetohydrodynamics (MHD), radiation MHD, electromagnetics, and incompressible hydrodynamics, satisfy Stoke's law type equations for the divergence-free evolution of vector fields. In this paper we present a full-fledged scheme for the second-order accurate, divergence-free evolution of vector fields on an adaptive mesh refinement (AMR) hierarchy. We focus here on adaptive mesh MHD. However, the scheme has applicability to the other systems of equations mentioned above. The scheme is based on making a significant advance in the divergence-free reconstruction of vector fields. In that sense, it complements the earlier work of D. S. Balsara and D. S. Spicer (1999, J. Comput. Phys. 7 , 270) where we discussed the divergence-free time-update of vector fields which satisfy Stoke's law type evolution equations. Our advance in divergence-free reconstruction of vector fields is such that it reduces to the total variation diminishing (TVD) property for one-dimensional evolution and yet goes beyond it in multiple dimensions. For that reason, it is extremely suitable for the construction of higher order Godunov schemes for MHD. Both the two-dimensional and three-dimensional reconstruction strategies are developed. A slight extension of the divergence-free reconstruction procedure yields a divergence-free prolongation strategy for prolonging magnetic fields on AMR hierarchies. Divergence-free restriction is also discussed. Because our work is based on an integral formulation, divergence-free restriction and prolongation can be carried out on AMR meshes with any integral refinement ratio, though we specialize the expressions for the most popular situation where the refinement ratio is two. Furthermore, we pay attention to the fact that in order to efficiently evolve the MHD equations on AMR hierarchies, the refined meshes must evolve in time with time steps that are a fraction of their parent mesh's time step. An electric field correction strategy is presented for use on AMR meshes. The electric field correction strategy helps preserve the divergence-free evolution of the magnetic field even when the time steps are subcycled on refined meshes. The above-mentioned innovations have been implemented in Balsara's RIEMANN framework for parallel, self-adaptive computational astrophysics, which supports both nonrelativistic and relativistic MHD. Several rigorous, three-dimensional AMR-MHD test problems with strong discontinuities have been run with the RIEMANN framework showing that the strategy works very well. In our AMR-MHD scheme, the adaptive mesh hierarchy can change in response to discontinuities that move rapidly with respect to the mesh. Time-step subcycling permits efficient processing of the AMR hierarchy. Our AMR-MHD scheme parallelizes very well as shown by Balsara and Norton [8].

[1]  Rainald Löhner,et al.  An adaptive finite element solver for transient problems with moving bodies , 1988 .

[2]  A Roe Scheme for Ideal MHD Equations on 2D Adaptively Refined Triangular Grids , 1999 .

[3]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[4]  P. Colella.  Multidimensional upwind methods for hyperbolic conservation laws , 1990 .

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

[6]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[7]  Isidore Rigoutsos,et al.  An algorithm for point clustering and grid generation , 1991, IEEE Trans. Syst. Man Cybern..

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

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

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

[11]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[12]  Dinshaw S. Balsara,et al.  Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods , 2001 .

[13]  Rainald Löhner,et al.  Adaptive remeshing for transient problems , 1989 .

[14]  William Gropp,et al.  Globalized Newton-Krylov-Schwarz Algorithms and Software for Parallel Implicit CFD , 2000, Int. J. High Perform. Comput. Appl..

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

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

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

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

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

[20]  P. Londrillo,et al.  High-Order Upwind Schemes for Multidimensional Magnetohydrodynamics , 1999, astro-ph/9910086.

[21]  Dinshaw S. Balsara.  The eigenstructure of the equations of radiation magnetohydrodynamics. , 1999 .

[22]  M. Brio,et al.  An upwind differencing scheme for the equations of ideal magnetohydrodynamics , 1988 .

[23]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[24]  A. Jameson.  Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings , 1991 .

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

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

[27]  Paul R. Woodward,et al.  An approximate Riemann solver for ideal magnetohydrodynamics , 1994 .

[28]  C. Richard DeVore,et al.  Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics , 1989 .

[29]  Dongsu Ryu,et al.  Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for multidimensional flow , 1995 .

[30]  Marsha Berger,et al.  Three-Dimensional Adaptive Mesh Refinement for Hyperbolic Conservation Laws , 1994, SIAM J. Sci. Comput..

[31]  F. Harlow,et al.  THE MAC METHOD-A COMPUTING TECHNIQUE FOR SOLVING VISCOUS, INCOMPRESSIBLE, TRANSIENT FLUID-FLOW PROBLEMS INVOLVING FREE SURFACES , 1965 .

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

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

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

[35]  Francesco Miniati,et al.  A Divergence-free Upwind Code for Multidimensional Magnetohydrodynamic Flows , 1998 .

[36]  Dinshaw S. Balsara.  LINEARIZED FORMULATION OF THE RIEMANN PROBLEM FOR RADIATION HYDRODYNAMICS , 1999 .

[37]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

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

[39]  Dinshaw S. Balsara,et al.  Highly parallel structured adaptive mesh refinement using parallel language-based approaches , 2001, Parallel Comput..

[40]  D. Balsara,et al.  The Evolution of Adiabatic Supernova Remnants in a Turbulent, Magnetized Medium , 2001, astro-ph/0107345.

[41]  P. Roe,et al.  Space-Time Methods for Hyperbolic Conservation Laws , 1998 .

[42]  A. Jeffrey,et al.  Nonlinear wave propagation , 1978 .

[43]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[44]  Dinshaw S. Balsara,et al.  Total Variation Diminishing Scheme for Adiabatic and Isothermal Magnetohydrodynamics , 1998 .

[45]  Paul R. Woodward,et al.  On the Divergence-free Condition and Conservation Laws in Numerical Simulations for Supersonic Magnetohydrodynamical Flows , 1998 .