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Approximate Riemann solvers and stable high-order finite volume schemes for multi-dimensional ideal MHD

by F. G. Fuchs and A. D. McMurray and S. Mishra and N. H. Risebro and K. Waagan

(Report number 2009-37)

Abstract
We design stable and high-order accurate finite volume schemes for the ideal MHD equations in multi-dimensions. We obtain excellent numerical stability due to some new elements in the algorithm. The schemes are based on three- and five-wave approximate Riemann solvers of the HLL-type, with the novelty that we allow a varying normal magnetic field. This is achieved by considering the semi-conservative Godunov-Powell form of the MHD equations. We show that it is important to discretize the Godunov-Powell source term in the right way, and that the HLL-type solvers naturally provide a stable upwind discretization. Second-order versions of the ENO- and WENO-type reconstructions are proposed, together with precise modifications necessary to preserve positive pressure and density. Extending the discrete source term to second order while maintaining stability requires non-standard techniques, which we present. The first- and second-order schemes are tested on a suite of numerical experiments demonstrating impressive numerical resolution as well as stability, even on every fine meshes.

Keywords:

@Techreport{FMMRW09_418,
  author = {F. G. Fuchs and A. D. McMurray and S. Mishra and N. H. Risebro and K. Waagan},
  title = {Approximate Riemann solvers and stable high-order finite volume schemes for multi-dimensional ideal MHD},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-37},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-37.pdf },
  year = {2009}
}

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