A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: theoretical framework

Bouchut, François; Klingenberg, Christian; Waagan, Knut

doi:10.1007/s00211-007-0108-8

Cited by 132 publications

(128 citation statements)

References 28 publications

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“…For hydrodynamics, these HLL type three wave solvers are denoted as HLLC solvers and were proposed in [43,44]. Three-wave HLL solvers for MHD have been designed in [8,21,25,30,31]. Further refinements on this theme include the design of five-wave solvers that also approximate Alfvén waves in [8,34].…”

Section: Survey Of Available Numerical Methodsmentioning

confidence: 99%

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Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Fuchs

McMurry

Mishra

et al. 2011

Commun. comput. phys.

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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 semiconservative 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 very fine meshes.

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Section: Survey Of Available Numerical Methodsmentioning

confidence: 99%

“…Three-wave HLL solvers for MHD have been designed in [8,21,25,30,31]. Further refinements on this theme include the design of five-wave solvers that also approximate Alfvén waves in [8,34]. Some of these HLL type solvers [8,21,34] are proved to preserve positive pressures and densities.…”

Section: Survey Of Available Numerical Methodsmentioning

confidence: 99%

Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Fuchs

McMurry

Mishra

et al. 2011

Commun. comput. phys.

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“…Other popular numerical fluxes used for the solution of MHD equations include linearized Roe solvers [9,36,37] and HLL type solvers [7,17,21,26,32]. Detailed comparisons of different solvers are performed in [18,31].…”

Section: Finite-volume Schemesmentioning

confidence: 99%

Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations

Mishra

Tadmor

2012

ESAIM: M2AN

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“…For some of them it is even possible to prove some entropy estimates (see [2]). The advantage of our approach is that it is very simple and general and that the resulting scheme has the same precision as the Roe or the Godunov solver.…”

Section: Application To Magnetohydrodynamicsmentioning

confidence: 99%

A simple parameter-free entropy correction for approximate Riemann solvers

Helluy

Hérard²,

Mathis

et al. 2010

Comptes Rendus. Mécanique

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We present here a simple and general non-parametrized entropy-fix for the computation of fluid flows involving sonic points in rarefaction waves. It enables to improve the stability and the accuracy of approximate Riemann solvers. It is also applied to MHD flows. L'approche proposée dans cette note consiste très simplementà localiser les interfaces entre deux volumes de contrôle séparant deux valeurs du nombre de Mach de part et d'autre de 1, dans les zones régulières, puisà remplacer localement sur cette interface le flux du solveur de Riemann approché par un flux de type Rusanov ([14]). La consistance du flux et la conservativité du schéma sont préservées, l'ordre de convergence global du schéma reste inchangé, et la précision globaleà maillage donné se trouve nettement améliorée par rapportà d'autres corrections entropiques paramétriques. Pour leséquations de la dynamique des gaz, il est alors toutà fait possible d'envisager des calculs en présence de très faibles densités et pressions (section 3) ; il en est de même pour la simulation de problèmes de magnétohydrodynamique ([1], et section 4). Cette correction estégalement très utile si l'on chercheà réaliser certaines simulations nécessitant une formulationà section variable ([11]). L'approche permet de traiter des problèmes de Riemann mettant en jeu des rapports de pression (ou densité) de l'ordre de 10 5 . Sur les maillages les plus grossiers, la perturbation résiduelle au niveau du point sonique est semblableà celle observée pour un schéma de Godunov exact. De nombreuses applicationsà d'autres systèmes deviennent alors envisageables ([12]). L'extension au cadre multidimensionnel est immédiate. To cite this article: Author, C. R. Mecanique xxx (2009). Résumé

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A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: theoretical framework

Cited by 132 publications

References 28 publications

Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Approximate Riemann Solvers and Robust High-Order Finite Volume Schemes for Multi-Dimensional Ideal MHD Equations

Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations

A simple parameter-free entropy correction for approximate Riemann solvers

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