A multiwave approximate Riemann solver for ideal MHD based on relaxation II: Numerical implementation with 3 and 5 waves

CNRS & LAMA Université Paris-Est Marne-la-Vallée 5 Boulevard Descartes Cité Descartes, Champs-sur-Marne 77454 Marne-la-Vallée cedex 2 France
Numerische Mathematik (Impact Factor: 1.61). 06/2010; 115(4):647-679. DOI: 10.1007/s00211-010-0289-4


In the first part of this work Bouchut etal. (J Comput Phys 108:7–41, 2007) we introduced an approximate Riemann solver for
one-dimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete
entropy inequalities, and to preserve positivity of density and internal energy. In this paper we consider the practical implementation,
and derive explicit wave speed estimates satisfying the stability conditions of Bouchut etal. (J Comput Phys 108:7–41, 2007).
We present a 3-wave solver that well resolves fast waves and material contacts, and a 5-wave solver that accurately resolves
the cases when two eigenvalues coincide. A full 7-wave solver, which is highly accurate on all types of waves, will be described
in a follow-up paper. We test the solvers on one-dimensional shock tube data and smooth shear waves.

Mathematics Subject Classification (2000)76M12-65M12-76W05

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    • "Such values have been found in [6, Proposition 2.18] for Euler equations and in [9] for MHD equations. We use here the approach of [9], that enables to treat negative pressures π. We make the following a priori choice of the relaxation speeds c l , c r , c l = h l s l + 3 2 h l (u l − u r ) + + (π r − π l ) + h l s l + h r s r , c r = h r s r + 3 2 h r (u l − u r ) + + (π l − π r ) + h l s l + h r s r , "
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    ABSTRACT: The shallow water magnetohydrodynamic system describes the thin layer evolution of the solar tachocline. It is obtained from the three dimensional incompressible magnetohydrodynamic system similarly as the classical shallow water system is obtained from the incompressible Navier-Stokes equations. The system is hyperbolic and has two additional waves with respect to the shallow water system, the Alfven waves. These are linearly degenerate, and thus do not generate dissipation. In the present work we introduce a 5-wave approximate Riemann solver for the shallow water magnetohydrodynamic system, that has the property to be non dis-sipative on Alfven waves. It is obtained by solving a relaxation system of Suliciu type, and is similar to HLLC type solvers. The solver is positive and entropy satisfying, ensuring its robustness. It has sharp wave speeds, and does not involve any iterative procedure.
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    • "The HLL5R scheme is the most efficient, however it requires slightly smaller time steps. The shorter time steps are not surprising, since the wave speeds of the HLL5R Riemann solver may be up to 10% larger than with the corresponding 3-wave solver in some configurations; see [5]. Consequently, there would be a roughly 10% reduction in time step size. "
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    ABSTRACT: The ideal MHD equations are a central model in astrophysics, and their solution relies upon stable numerical schemes. We present an implementation of a new method, which possesses excellent stability properties. Numerical tests demonstrate that the theoretical stability properties are valid in practice with negligible compromises to accuracy. The result is a highly robust scheme with state-of-the-art efficiency. The scheme's robustness is due to entropy stability, positivity and properly discretised Powell terms. The implementation takes the form of a modification of the MHD module in the FLASH code, an adaptive mesh refinement code. We compare the new scheme with the standard FLASH implementation for MHD. Results show comparable accuracy to standard FLASH with the Roe solver, but highly improved efficiency and stability, particularly for high Mach number flows and low plasma beta. The tests include 1D shock tubes, 2D instabilities and highly supersonic, 3D turbulence. We consider turbulent flows with RMS sonic Mach numbers up to 10, typical of gas flows in the interstellar medium. We investigate both strong initial magnetic fields and magnetic field amplification by the turbulent dynamo from extremely high plasma beta. The energy spectra show a reasonable decrease in dissipation with grid refinement, and at a resolution of 512^3 grid cells we identify a narrow inertial range with the expected power-law scaling. The turbulent dynamo exhibits exponential growth of magnetic pressure, with the growth rate twice as high from solenoidal forcing than from compressive forcing. Two versions of the new scheme are presented, using relaxation-based 3-wave and 5-wave approximate Riemann solvers, respectively. The 5-wave solver is more accurate in some cases, and its computational cost is close to the 3-wave solver.
    Journal of Computational Physics 01/2011; 230(9-230). DOI:10.1016/j.jcp.2011.01.026 · 2.43 Impact Factor
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    • "The exact solution can be found in [5]. The resolution was h = 0.005, and ρ is plotted at t = 0.1. "
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    ABSTRACT: We have developed approximate Riemann solvers for ideal MHD equations based on a relaxation approach in [4],[5]. These lead to entropy consistent solutions with good properties like guaranteed positive density. We describe the extension to higher order and multiple space dimensions. Finally we show our implementation of all this into the astrophysics code FLASH.
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