Open Access

ARTICLE

High-Order Unstructured One-Step PNPMSchemes for the Viscous and Resistive MHD Equations

M. Dumbser¹, D.S. Balsara²

1 Laboratory of Applied Mathematics, University of Trento, Via Mesiano 77, I-38100 Trento, Italy.
2 University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA.

Computer Modeling in Engineering & Sciences 2009, 54(3), 301-334. https://doi.org/10.3970/cmes.2009.054.301

Abstract

In this article we use the new, unified framework of high order one-step P_NP_M schemes recently proposed for inviscid hyperbolic conservation laws by Dumbser, Balsara, Toro, and Munz (2008) in order to solve the viscous and resistive magnetohydrodynamics (MHD) equations in two and three space dimensions on unstructured triangular and tetrahedral meshes. The P_NP_M framework uses piecewise polynomials of degree N to represent data in each cell and piecewise polynomials of degree M ≥ N to compute the fluxes and source terms. This new general machinery contains usual high order finite volume schemes (N = 0) and discontinuous Galerkin finite element methods (N = M) just as special cases of a more general unified class of numerical methods. The new time discretization approach used in our schemes is of the one-step type and is obtained by a local space-time Galerkin predictor-corrector method, where in the predictor step a local weak form of the governing PDE is solved in the small. The formal order of accuracy of our one-step time discretization automatically matches the spatial discretization order.
We perform a numerical convergence study for an unsteady test case on unstructured meshes with P_NP_M methods from third to eighth order of accuracy in space and time and show detailed CPU times to allow the reader a critical performance assessment. We present numerical results for the viscous Orszag-Tang vortex system, for the first problem of Stokes and for a steady laminar boundary layer at high Reynolds number. We furthermore compute the unsteady flow past a circular cylinder and the Kelvin-Helmholtz instability in two and three space dimensions.

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