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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Available from: François Bouchut, Oct 07, 2015
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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/ · 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.
    Acta Mathematica Scientia 03/2010; 30(2):621-632. DOI:10.1016/S0252-9602(10)60065-3 · 0.74 Impact Factor
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    ABSTRACT: We present a new HLL-type approximate Riemann solver that aims at capturing any isolated discontinuity without necessitating extensive characteristic analysis of governing partial differential equations. This property is especially attractive for complex hyperbolic systems with more than two equations. Following Linde’s approach [6], we introduce a generic middle wave into the classical two-state HLL solver. The property of this third wave is typified by the way of a “strength indicator” that is derived from polynomial considerations. The polynomial that constitutes the basis of the procedure is made non-oscillatory by an adapted fourth-order WENO algorithm (CWENO4). This algorithm makes it possible to derive an expression for the strength indicator. According to the size of this latter parameter, the resulting solver (HLL-RH), either computes the multi-dimensional Rankine-Hugoniot equations if an isolated discontinuity appears in the Riemann fan, or asymptotically tends towards the two-state HLL solver if the solution is locally smooth. The asymptotic version of the HLL-RH solver is demonstrated to be positively conservative and entropy satisfying in its first-order multi-dimensional form provided that a relevant and not too restrictive CFL condition is considered; specific limitations of the conservative increments of the numerical solution and a suited entropy condition enable to maintain these properties in its high-order version.With a monotonicity-preserving algorithm for the time integration, the numerical method so generated, is third order in time and fourth-order accurate in space for the smooth part of the solution; moreover, the scheme is stable and accurate when capturing a shock wave, whatever the complexity of the underlying differential system.Extensive numerical tests for the one- and two-dimensional Euler equation of gas dynamics and comparisons with classical Godunov-type methods help to point out the potentialities and insufficiencies of the method.
    Computers & Fluids 09/2012; 68:219–243. DOI:10.1016/j.compfluid.2012.07.001 · 1.62 Impact Factor
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