A Variant of Van Leer's Method for Multidimensional Systems of Conservation Laws

Perthame, Benoı̂t; Qiu, Youchun

doi:10.1006/jcph.1994.1107

Cited by 38 publications

(58 citation statements)

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“…We skip the proof of Theorem 4, which is a straightforward extension of that in §1. The only new point is to introduce, following [8] Then the exponents in (29) are uniquely recovered by the requirement that, (fgy~2)x^y~x^ being homogeneous to Sl{...}, the minimum in Lemma 2, with the constraint fgy~2 < 37-1, be achieved for our choice of x> <P m (29).…”

Section: The 2d Casementioning

confidence: 99%

Maximum principle on the entropy and second-order kinetic schemes

Khobalatte¹,

Perthame²

1994

Math. Comp.

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Abstract.We consider kinetic schemes for the multidimensional inviscid gas dynamics equations (compressible Euler equations). We prove that the discrete maximum principle holds for the specific entropy. This fixes the choice of the equilibrium functions necessary for kinetic schemes. We use this property to perform a second-order oscillation-free scheme, where only one slope limitation (for three conserved quantities in 1D) is necessary. Numerical results exhibit stability and strong convergence of the scheme.

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Section: The 2d Casementioning

confidence: 99%

Maximum principle on the entropy and second-order kinetic schemes

Khobalatte¹,

Perthame²

1994

Math. Comp.

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“…The well-known Godunov scheme [13,20], but also the Lax-Friedrichs scheme (see Lax [18] or Tadmor [24] but also [20,26]), the kinetic scheme as proposed by Perthame (see KhobalattePerthame [16], Perthame [21] and Perthame-Qiu [23]) or the Suliciu relaxation scheme (Bouchut [5], Coquel-Perthame [10], Berthon [4] for instance), satisfy the positiveness of the density and the internal energy, a set of discrete entropy inequalities (1.5) and a discrete formulation of the minimum principle on the specific entropy (1.7). In the sequel, the scheme (1.8) will be assumed to satisfy the following properties:…”

Section: Introduction the Present Work Is Devoted To The Numerical Amentioning

confidence: 99%

“…The Godunov-type scheme certainly denotes the main class of second-order scheme. It was largely studied in the last twenty years [2,3,6,8,9,15,16,19,21,22,23]. Unfortunately, these procedures are, in general, based on the generalized Riemann problem (see Ben-Artzi-Falcovitz [3] or Bourgeade-LeFloch-Raviart [7]).…”

Section: Introduction the Present Work Is Devoted To The Numerical Amentioning

confidence: 99%

“…In fact, the conservative variables W are not the most used and several papers (see [16,23,12]) consider a piecewise linear reconstruction on a relevant change of variables in the form:…”

Section: Introduction the Present Work Is Devoted To The Numerical Amentioning

confidence: 99%

“…However, KhobalattePerthame [16] and Perthame-Qiu [23] exhibit (conservative) slope limitations in order to preserve the invariant region; namely the positiveness of both density and internal energy. This result is established in the framework of the kinetic scheme with a relevant like CFL restriction.…”

Section: Introduction the Present Work Is Devoted To The Numerical Amentioning

confidence: 99%

See 2 more Smart Citations

Stability of the MUSCL Schemes for the Euler Equations

Berthon¹

2005

Communications in Mathematical Sciences

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Abstract. The second-order Van-Leer MUSCL schemes are actually one of the most popular high order scheme for fluid dynamic computations. In the frame work of the Euler equations, we introduce a new slope limitation procedure to enforce the scheme to preserve the invariant region: namely the positiveness of both density and pressure as soon as the associated first order scheme does it. In addition, we obtain a second-order minimum principle on the specific entropy and secondorder entropy inequalities. This new limitation is developed in the general framework of the MUSCL schemes and the choice of the numerical flux functions remains free. The proposed slope limitation can be applied to any change of variables and we do not impose the use of conservative variables in the piecewise linear reconstruction. Several examples are given in the framework of the primitive variables. Numerical 1D and 2D results are performed using several finite volume methods.

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Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation

Wu,

Trask,

Chan

2024

Numerical Methods Partial

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High order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles.

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A Variant of Van Leer's Method for Multidimensional Systems of Conservation Laws

Cited by 38 publications

References 0 publications

Maximum principle on the entropy and second-order kinetic schemes

Maximum principle on the entropy and second-order kinetic schemes

Stability of the MUSCL Schemes for the Euler Equations

Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation

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