Polynomial upwind schemes for hyperbolic systems

Degond, Pierre; Peyrard, Pierre-François; Russo, Giovanni; Villedieu, Philippe

doi:10.1016/s0764-4442(99)80194-3

Cited by 57 publications

(77 citation statements)

References 7 publications

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“…The goal of this work is to introduce a new viscosity matrix leading to a numerical scheme whose accuracy is close to that of the Roe scheme and whose computational cost is similar to the FORCE o GFORCE methods. In order to introduce such as viscosity matrix, let us first observe that the matrix |A Φ (w L , w R )| can be rewritten as follows (see [16]): 19) where α j , j = 0, · · · , 3 are defined in terms of the eigenvalues…”

Section: Numerical Schemesmentioning

confidence: 99%

“…This Riemann solver is based on a suitable decomposition of a Roe matrix (see [27]) by means of a parabolic viscosity matrix (see [16]) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L ∞ -stable, well-balanced, and it doesn't require an entropy-fix technique.…”

Section: Introductionmentioning

confidence: 99%

“…In this work, we present an approximate Riemann solver that uses some information concerning the internal eigenvalues without requiring the computation of the complete eigenstructure. The definition of the method, which is based on a suitable decomposition of a Roe matrix (see [27]) by means of a parabolic viscosity matrix (see [16]) is presented in the next section. The well-balanced properties of the numerical scheme are studied in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

2011

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The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see [14]). This Riemann solver is based on a suitable decomposition of a Roe matrix (see [27]) by means of a parabolic viscosity matrix (see [16]) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L ∞ -stable, well-balanced, and it doesn't require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods.

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Section: Numerical Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

2011

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“…1, 2 and 3, we present the profiles of the density, the mean velocity and the temperature, as functions of the scaled space. These figures display the results obtained by the full BGK equation (discretized by (3.1)), by the Navier-Stokes equations (discretized following [16]) and by the hybrid model, for different values of R (R = 0.5, 2 and 4).…”

Section: Stationary Shock Wave Without Electric Field 411 Plasma Comentioning

confidence: 99%

“…Usually, the Fokker-Planck-Landau (FPL) collision operator is the common collision model in plasma physics. Classically, the FPL operator is obtained as a remedy to the loss of finiteness of the Boltzmann operator for long-range Coulomb interaction (see [16,21]). It takes the form of a nonlinear partial integro-differential operator whose direct numerical implementation is very expensive and difficult.…”

Section: Introductionmentioning

confidence: 99%

A hybrid kinetic–fluid model for solving the Vlasov–BGK equation

Crouseilles¹,

Degond²,

Lemou³

2005

Journal of Computational Physics

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Our purpose is to derive a model for charged particles which combines a kinetic description of the fast particles with a fluid description of the slow ones. In a previous paper, a similar model was derived from a kinetic BGK equation that uses a constant relaxation time and does not include the effect of an electric field. In this paper, we consider a more general kinetic equation including an electric field and a varying relaxation time. Fast particles will be described through a collisional kinetic equation of Vlasov-BGK type while thermal particles will be modeled by a hydrodynamic model. Then, we construct a numerical scheme for this model and validate the approach by presenting various numerical tests.

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Relation between PVM schemes and simple Riemann solvers

Luna

Díáz²,

Parés³

2014

Numerical Methods Partial

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Approximate Riemann solvers (ARS) and polynomial viscosity matrix (PVM) methods constitute two general frameworks to derive numerical schemes for hyperbolic systems of Partial Differential Equations (PDE's). In this work, the relation between these two frameworks is analyzed: we show that every PVM method can be interpreted in terms of an approximate Riemann solver provided that it is based on a polynomial that interpolates the absolute value function at some points. Furthermore, the converse is true provided that the ARS satisfies a technical property to be specified. Besides its theoretical interest, this relation provides a useful tool to investigate the properties of some well-known numerical methods that are particular cases of PVM methods, as the analysis of some properties is easier for ARS methods. We illustrate this usefulness by analyzing the positivity-preservation property of some well-known numerical methods for the shallow water system.

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Polynomial upwind schemes for hyperbolic systems

Cited by 57 publications

References 7 publications

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

A hybrid kinetic–fluid model for solving the Vlasov–BGK equation

Relation between PVM schemes and simple Riemann solvers

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