Multidimensional upwind schemes based on fluctuation-splitting for systems of conservation laws

Deconinck, H.; Paillère, H.; Struijs, R.; Roe, Philip L.

doi:10.1007/bf00350091

Cited by 63 publications

(28 citation statements)

References 5 publications

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“…The flow is assumed to be one-dimensional, but not necessarily in a direction related to the element. This produced results with much crisper discontinuities than those from grid-aligned upwinding [8]. The generalization of these methods, first to systems of equations, then to unsteady flows, and eventually to viscous flows [1] has however followed a path that I find slightly unconvincing, despite its very real success.…”

Section: What Else Could There Be?mentioning

confidence: 99%

Is Discontinuous Reconstruction Really a Good Idea?

Roe

2017

J Sci Comput

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It has been almost automatically assumed for a quarter century that the numerical solution of hyperbolic conservation laws is best accomplished by making a reconstruction of the initial data that is only piecewise continuous. The effect of the discontinuities is taken into account by means of Riemann solvers. This strategy has enjoyed great practical success but introduces only one-dimensional physics as a guide to the discretization of multidimensional problems. This article points out some of the resulting defects and proposes an alternative viewpoint. The chief novelty of the new "Active Flux" method, apart from the elimination of discontinuities, is the division into advective and acoustic disturbances, with acoustics being handled by exploiting classical solutions to the scalar wave equation. Results for a standard test problem for the compressible Euler equations indicate that an order of magnitude increase in cost effectiveness may be possible by following this approach, compared to the traditional one.

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Section: What Else Could There Be?mentioning

confidence: 99%

Is Discontinuous Reconstruction Really a Good Idea?

Roe

2017

J Sci Comput

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“…A substantial body of current research is directed toward the implementation of truly multidimensional upwind schemes in which the upwind biasing is determined by properties of the flow rather than the mesh (Hirsch, Lacol and Deconinck, 1987;Powell and van Leer, 1989;Van Leer, 1993;Deconinck et al, 1993;Sidilkover, 1995). A thorough review is given by Paillère and Deconinck (1995).…”

Section: Multidimensional Shock Capturing Schemesmentioning

confidence: 99%

Aerodynamics

Jameson

2004

Encyclopedia of Computational Mechanics

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Computational methods are now pervasive in the science of aerodynamics. Because previously existing numerical methods proved inadequate for fluid flow simulations, the emergence of computational fluid dynamics (CFD) as a distinct discipline has sparked the development of an entirely new class of algorithms and a supporting body of theory, which are the main theme of this article. After a review of mathematical models of fluid flow, methods for solving the transonic potential flow equation (of mixed type) are examined. The central part of the article discusses the formulation and implementation of shock‐capturing schemes for the Euler and Navier–Stokes equations. The article next discusses the merits of the contrasting approaches of finite difference, finite volume, and finite element methods for the treatment of flows in complex geometric domains, together with the treatment of boundary conditions. A detailed presentation of explicit and implicit time‐stepping schemes for both steady and unsteady flows precedes a discussion of preconditioning and multigrid procedures for accelerating the convergence of steady state calculations. Dual time‐stepping schemes are recommended for the simulation of unsteady flow. In order to realize the potential benefits of CFD, it is essential to move beyond simulation to aerodynamic (and ultimately multidisciplinary) optimization. The article concludes with a discussion of aerodynamic shape optimization via control theory.

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“…The fluctuation splitting schemes, also called residual distribution schemes, have been largely presented in the literature, for instance in [5] or in [9].…”

Section: The Fluctuation Splitting Formalismmentioning

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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Multidimensional upwind schemes based on fluctuation-splitting for systems of conservation laws

Abstract: A class of truly multidimensional upwind schemes for the computation of inviscid compressible flows is presented here, applicable to unstructured cell-vertex grids. These methods use very compact stencils and produce sharp resolution of discontinuities with no overshoots.

Cited by 63 publications

References 5 publications

Is Discontinuous Reconstruction Really a Good Idea?

Is Discontinuous Reconstruction Really a Good Idea?

Aerodynamics

A multidimensional fluctuation splitting scheme for the three dimensional Euler equations

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