Progress on multidimensional upwind Euler solvers for unstructured grids

In this paper we consider the solution of hyperbolic conservation laws on moving meshes by means of an Arbitrary Lagrangian Eulerian (ALE) formulation of the Runge-Kutta Residual Distribution (RD) schemes of Ricchiuto and Abgrall (J Comput Phys 229(16):5653-5691, 2010). Up to the authors knowledge, the problem of recasting RD schemes into ALE framework has been solved with first order explicit schemes and with second order implicit schemes. Our resulting scheme is explicit and second order accurate when computing discontinuous solutions.

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“…Multidimensional upwinding, as introduced by Roe, Deconinck and collaborators (see e.g. [18]), corresponds to the condition…”

Section: Upwindingmentioning

confidence: 99%

An ALE Formulation for Explicit Runge–Kutta Residual Distribution

2014

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“…produces the PSI scheme [11], but a more flexible alternative is to blend two schemes, such as N and LDA, i.e.…”

Section: Space-time Fluctuation Distributionmentioning

confidence: 99%

Discontinuous fluctuation distribution

Hubbard

2008

Journal of Computational Physics

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This paper proposes an approach to the approximation of timedependent hyperbolic conservation laws which is both second order accurate in space and time (for any sufficiently smooth solution profile, even one containing turning points) and free of spurious oscillations for any time-step. The numerical algorithm is based on the concept of fluctuation distribution, applied on a space-time mesh of triangular prisms, for which second order accurate schemes already exist which are oscillation-free if the time-step satisfies a CFL-type constraint. This restriction is lifted here by combining the concept of a two-layer scheme with a representation of the solution which is allowed to be discontinuous in time. Numerical results are presented in two space dimensions, using unstructured meshes of space-time triangular prisms, for the scalar advection equation, Burgers' equation and the Euler equations of gasdynamics.Key-words: conservation laws, positivity, stability, large time steps, residual distribution, second order accuracy * University of Leeds † INRIA Bordeaux -Sud-Ouest Discontinuous residual distribution : A route to unconditional positivity and high order accuracyRésumé : This paper proposes an approach to the approximation of timedependent hyperbolic conservation laws which is both second order accurate in space and time (for any sufficiently smooth solution profile, even one containing turning points) and free of spurious oscillations for any time-step. The numerical algorithm is based on the concept of fluctuation distribution, applied on a space-time mesh of triangular prisms, for which second order accurate schemes already exist which are oscillation-free if the time-step satisfies a CFL-type constraint. This restriction is lifted here by combining the concept of a two-layer scheme with a representation of the solution which is allowed to be discontinuous in time. Numerical results are presented in two space dimensions, using unstructured meshes of space-time triangular prisms, for the scalar advection equation, Burgers' equation and the Euler equations of gasdynamics.

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“…The extension of the fluctuation splitting method to non linear conservation laws is largely discussed in [9] and the conclusion is that the linearization L Z [6,20] has to be used as far as we have to get a constant approximation of whatever quantity.…”

Section: The Roe Linearizationmentioning

confidence: 99%

A multidimensional fluctuation splitting scheme for the three dimensional Euler equations

Bastin

Rogé²

1999

ESAIM: M2AN

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Abstract. The fluctuation splitting schemes were introduced by Roe in the beginning of the 80's and have been then developed since then, essentially thanks to Deconinck. In this paper, the fluctuation splitting schemes formalism is recalled. Then, the hyperbolic/elliptic decomposition of the three dimensional Euler equations is presented. This decomposition leads to an acoustic subsystem and two scalar advection equations, one of them being the entropy advection. Thanks to this decomposition, the two scalar equations are treated with the well known PSI scalar fluctuation splitting scheme, and the acoustic subsystem is treated with the Lax Wendroff matrix fluctuation splitting scheme. An additional viscous term is introduced in order to reduce the oscillatory behavior of the Lax Wendroff scheme. An implicit form leads to a robust scheme which enables computations over a large range of Mach number. This fluctuation splitting scheme, called the Lax Wendroff -PSI scheme, produces little spurious entropy, thus leading to accurate drag predictions. Numerical results obtained with this Lax Wendroff PSI scheme are presented and compared to a reference Euler code, based on a Lax Wendroff scheme.Résumé. Les schémas distributifs sont apparus au début des années 80 grâceà Roe et ont connu de nombreux développements jusqu'à aujourd'hui, notamment sous l'impulsion de Deconinck. Dans cet article, nous rappelons le formalisme des schémas distributifs avant de présenter la décomposition hyperbolique elliptique deséquations d'Euler tridimensionnelles. Cette décomposition permet d'obtenir un sous-système acoustique et deuxéquations d'advection scalaire, dont l'advection de l'entropie. Grâcè a cette décomposition, les deuxéquations scalaires sont traitées par le schéma PSI, célèbre schéma distributif scalaire, et le sous-système acoustique est traitéà l'aide du schéma distributif Lax Wendroff matriciel. Une viscosité artificielle est utilisée pour réduire le caractère oscillant du schéma de Lax Wendroff. Après un important travail d'implicitation, on obtient un code robuste permettant des calculs précis dans une large zone de nombre de Mach. Le schéma distributif ainsi obtenu, le schéma Lax Wendroff -PSI, permet de réduire l'entropie numérique et donc d'obtenir des calculs de traînées précis. Enfin, des résultats numériques obtenusà l'aide de ce schéma sont présentés et comparés avec un code Euler de référence, utilisant une approche de type Lax Wendroff.

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Progress on multidimensional upwind Euler solvers for unstructured grids

Cited by 37 publications

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An ALE Formulation for Explicit Runge–Kutta Residual Distribution

An ALE Formulation for Explicit Runge–Kutta Residual Distribution

Discontinuous fluctuation distribution

A multidimensional fluctuation splitting scheme for the three dimensional Euler equations

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