A Simple Finite Difference Scheme for Multidimensional Magnetohydrodynamical Equations

Wenlong Dai, Paul R. Woodward

Research output: Contribution to journalArticlepeer-review

132 Scopus citations


An approximate MHD Riemann solver, an approach to maintain the divergence-free condition of magnetic field, and a finite difference scheme for multidimensional magnetohydrodynamical (MHD) equations are proposed in this paper. The approximate MHD Riemann solver is based on characteristic formulations. Both the conservation laws for mass, momentum, energy, and magnetic field, and the divergence-free condition of the magnetic field are exactly satisfied in the proposed scheme. The scheme does not involve any Poisson solver and is second-order accurate in both space and time. The correctness and robustness of the scheme are shown through numerical examples. The approach proposed in this paper to maintain the divergence-free condition may be applied to other dimensionally split and unsplit Godunov schemes for MHD flows.

Original languageEnglish (US)
Pages (from-to)331-369
Number of pages39
JournalJournal of Computational Physics
Issue number2
StatePublished - May 20 1998

Bibliographical note

Funding Information:
One of the authors, W. Dai, thanks Dr. David Porter for the discussion about the convective instability. The work presented here has been supported by the Department of Energy through Grants DE-FG02-87ER25035 and DE-FG02-94ER25207, by the National Science Foundation through Grant ASC-9309829, by NASA through Grant USRA/5555-23/NASA, and by the University of Minnesota through its Minnesota Supercomputer Institute.


  • Finite difference
  • Godunov scheme
  • Hyperbolic system
  • MHD simulation


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