Maximum Principle on the Entropy and Second-Order Kinetic Schemes

Khobalatte, Brahim; Perthame, Benôıt

doi:10.2307/2153399

Cited by 30 publications

(54 citation statements)

References 16 publications

(15 reference statements)

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“…TVD type high order Runge-Kutta time discretization (Shu and Osher [11]) will keep the validity of Theorem 1, for convexity reasons, as it was shown in [7].…”

Section: Remarkmentioning

confidence: 89%

“…It will also obviously satisfy the positivity property. Both of these Lax-Friedrichs schemes in addition satisfy the maximum principle on the specific entropy (Tadmor [13], [7]), but they are not multidimensional schemes in the sense that they are not rotation invariant.…”

Section: Appendix: Positivity Of the Lax-friedrichs Schemementioning

confidence: 99%

“…This is at least a first step towards entropy inequalities which require the density and pressure to be positive in order to define the physical entropy ρ ln(ρ γ /p) for a perfect gas. Another possible a priori bound is the maximum principle for the specific entropy (Tadmor [13]), which seems extremely difficult to preserve for second and higher order schemes (Khobalatte and Perthame [7]). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On positivity preserving finite volume schemes for Euler equations

Perthame

Shu

1996

Numerische Mathematik

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211

143

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We consider the positivity preserving property of first and higher order finite volume schemes for one and two dimensional Euler equations of gas dynamics. A general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on an exact or approximate Riemann solver and the reconstruction are both positivity preserving. Appropriate limitation to achieve a high order positivity preserving reconstruction is described.

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“…TVD type high order Runge-Kutta time discretization (Shu and Osher [11]) will keep the validity of Theorem 1, for convexity reasons, as it was shown in [7].…”

Section: Remarkmentioning

confidence: 89%

Section: Appendix: Positivity Of the Lax-friedrichs Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On positivity preserving finite volume schemes for Euler equations

Perthame

Shu

1996

Numerische Mathematik

Self Cite

211

143

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“…Linde and Roe [9] discussed the conditions for a second-order multidimensional MUSCL-type scheme to remain positively conservative. Perthame [11] discussed the positivity property for the kinetic scheme (See also [8]). Perthame and Shu [12] discussed the positivity preserving finite volume methods for the compressible Euler equations in general.…”

Section: Introductionmentioning

confidence: 99%

Pseudoparticle representation and positivity analysis of explicit and implicit Steger-Warming FVS schemes

Tang¹,

Xu²

2001

Z. angew. Math. Phys.

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This paper is about the pseudo-particle representation and the positivity analysis of an explicit and an implicit Steger and Warming's flux vector splitting (FVS) scheme for the compressible Euler equations. The positivity proof is based on the motion of pseudo-particles. For the explicit scheme, it shows that the density and the internal energy could keep non-negative values under the CFL-like condition for the Steger-Warming FVS scheme once the initial gas stays in a physically realizable state. For the implicit method, under a stronger CFL-like condition, the positivity property can also be preserved. Mathematics Subject Classification (2000). 35L65, 65M06, 76M20, 76K05.

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“…The derivation of slope limitations for the Euler system based on the specific entropy can be traced back to the paper by Khobalatte-Perthame [24]. See also Osher-Chakravarthy [34].…”

Section: Introductionmentioning

confidence: 99%

An entropy satisfying MUSCL scheme for systems of conservation laws

Coquel

LeFloch

1996

Numerische Mathematik

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For the high-order numerical approximation of hyperbolic systems of conservation laws, we propose to use as a building principle an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by Harten for scalar equations. Based on this new criterion, we derive entropy diminishing projections that ensure, both, the second order of accuracy and all of the classical discrete entropy inequalities. The resulting scheme is a nonlinear version of the classical Van Leer's MUSCL scheme. Strong convergence of this second order, entropy satisfying scheme is proved for systems of two equations. Numerical tests demonstrate the interest of our theory.

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Maximum Principle on the Entropy and Second-Order Kinetic Schemes

Cited by 30 publications

References 16 publications

On positivity preserving finite volume schemes for Euler equations

On positivity preserving finite volume schemes for Euler equations

Pseudoparticle representation and positivity analysis of explicit and implicit Steger-Warming FVS schemes

An entropy satisfying MUSCL scheme for systems of conservation laws

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