A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes

LeVeque, Randall J.; Pelanti, Marica

doi:10.1006/jcph.2001.6838

Cited by 77 publications

(88 citation statements)

References 45 publications

(53 reference statements)

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“…Our approach was in particular suggested by the recent relaxation solver of Berthon and Marche [21], but the main ideas follow primarily the work of Jin and Xin [29] and Suliciu [18,19]. Other related works are for instance [30,31,23]. We refer in particular to the monograph [20] and the bibliography therein.…”

Section: Relaxation Methods For the Single-phase Shallow Flow Modelmentioning

confidence: 99%

“…When the algorithm above is used, the Riemann solution of the relaxation system results in the definition of an approximate Riemann solution q rel RS ðx=t; q ' ; q r Þ for the original system, see [23,20]. The resulting numerical approximation method for (1) is a Godunov-type scheme in the sense of Harten-Lax-van Leer (44) associated to a function q rel RS ðx=t; q ' ; q r Þ [20].…”

Section: Relaxation Methods For the Single-phase Shallow Flow Modelmentioning

confidence: 99%

“…Other approximate solvers able to treat efficiently vacuum states have been developed by means of relaxation strategies, such as Suliciu's solver [18][19][20], and the recent method of Berthon-Marche [21]. Among other methods, let us mention the augmented four-wave Riemann solver of George [22], which is related to the class of relaxation solvers of LeVeque and Pelanti [23], and the modified Roe method (MRoe) of Castro, Parés and co-workers [24,25]. This MRoe method however is not rigorously positivity preserving, and it may need a restriction on the CFL number to avoid negative water depths.…”

Section: Introductionmentioning

confidence: 99%

“…Both our relaxation model and the one in [21] are based on the pioneering idea of Jin and Xin [29] of approximating the original model equations via a new system that is easier to solve (see also e.g. [30,31,23]). The particular feature that we have borrowed from Berthon and Marche [21] is the formulation of a relaxation system with linear degeneracy in all the characteristic fields, a property obtained by a special decoupling of the linear equations governing the relaxation variables from the remaining non-linear equations of the relaxation model.…”

Section: Introductionmentioning

confidence: 99%

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A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

Pelanti

Bouchut

Mangeney

2011

Journal of Computational Physics

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a b s t r a c tWe present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that cannot be efficiently addressed by existing solvers. In particular, we are concerned with ensuring a robust treatment of dry bed states. The relaxation system used by the proposed solver is formulated by introducing auxiliary variables that replace the momenta in the spatial gradients of the original model systems. The resulting relaxation solver is related to Roe solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe's Riemann solution. The relaxation solver has the advantage of a certain degree of freedom in the specification of the wave structure through the choice of the relaxation parameters. This flexibility can be exploited to handle robustly vacuum states, which is a well known difficulty of standard Roe's method, while maintaining Roe's low diffusivity. For the single-phase model positivity of flow height is rigorously preserved. For the two-phase model positivity of volume fractions in general is not ensured, and a suitable restriction on the CFL number might be needed. Nonetheless, numerical experiments suggest that the proposed two-phase flow solver efficiently models wet/dry fronts and vacuum formation for a large range of flow conditions.As a corollary of our study, we show that for single-phase shallow flow equations the relaxation solver is formally equivalent to the VFRoe solver with conservative variables of Gallouët and Masella [T. Gallouët, J.-M. Masella, Un schéma de Godunov approché C.R. Acad. Sci. Paris, Série I, 323 (1996) 77-84]. The relaxation interpretation allows establishing positivity conditions for this VFRoe method.

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Section: Relaxation Methods For the Single-phase Shallow Flow Modelmentioning

confidence: 99%

Section: Relaxation Methods For the Single-phase Shallow Flow Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

Pelanti

Bouchut

Mangeney

2011

Journal of Computational Physics

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“…Relaxation approximations for conservation laws were deeply investigated in [29,38,31] and extended to the diffusive case of parabolic equations in [28,23,37]; high order numerical schemes were introduced in [11,12,44]. Moreover, relaxation models based on the Bhatnagar-Gross-Krook (BGK) kinetic approach were developed in [30,2].…”

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confidence: 99%

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Cavalli¹,

Naldi²,

Perugia³

2012

SIAM J. Sci. Comput.

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Abstract. In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter ε. When ε → 0 + , the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing ε yields a consistent discretization of the original PDE. The numerical schemes obtained with this procedure do not require solving implicit nonlinear problems and possess the robustness of upwind discretizations. The proposed approximations are based on a discontinuous Galerkin method in space and on suitable implicitexplicit integration in time. Then, in principle, we can achieve any order of accuracy and obtain stable solutions, even when the diffusion equation becomes degenerate and solution singularities develop. Moreover, when needed, we can easily incorporate slope limiters within our schemes in order to handle spurious oscillatory phenomena. Some preliminary theoretical results are given, along with several numerical tests in one and two space dimensions, both for linear and nonlinear diffusion problems, including a degenerate diffusion equation, that provide numerical evidence of the properties of the presented approach. Key words. discontinuous Galerkin method, relaxation models, nonlinear diffusion AMS subject classifications. 65M60, 65M12, 35K55DOI. 10.1137/110827752 1. Introduction. Linear and nonlinear diffusion equations come from a variety of diffusion phenomena widely appearing in nature. They are suggested as mathematical models in many fields, such as filtration, phase transition, biochemistry, image analysis, and dynamics of biological groups. In the nonlinear case, the classical solutions to many of these PDEs fail to exist in finite time, even if the initial data are smooth. In such cases, suitable criteria have been introduced, which allow one to select physically relevant weak solutions beyond the singularity time.Recently, relaxation approximations to such PDEs have been introduced. These methods are based on replacing the equation by a semilinear hyperbolic system with stiff relaxation terms, tuned by a relaxation parameter ε. When ε → 0 + , the solution of this system "relaxes" onto the solution of the original PDE. Thus a consistent discretization of the relaxation system for ε = 0 yields a consistent discretization of the original PDE, as can be seen, for instance, in [29] and [2]. The advantage of this procedure is that the numerical scheme obtained in this fashion does not need approximate Riemann solvers for the convective term but possesses the robustness of upwind discretizations. Moreover, the complexity introduced by replacing the original

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Finite Volume Methods: Foundation and Analysis

Barth

Herbin

Ohlberger

2017

Encyclopedia of Computational Mechanics Second Edition

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Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, porous media flow, meteorology, electromagnetics, models of biological processes, semi-conductor device simulation and many other engineering areas governed by conservative systems that can be written in integral control volume form.This article reviews elements of the foundation and analysis of modern finite volume methods.The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for non-oscillatory discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, E-fluxes, TVD discretization, nonoscillatory reconstruction, slope limiters, positive coefficient schemes, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of dffision terms and the extension to systems of nonlinear conservation laws.

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A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes

Cited by 77 publications

References 45 publications

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Finite Volume Methods: Foundation and Analysis

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