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Why many theories of shock waves are necessary. Convergence error in formally pathconsistent schemes
, 2008
"... ..."
2012), Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems
 Proc
"... Abstract. We consider a class of nonconservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy. We formulate a wellposed Riemann problem by supplementing it with a kinetic relation, that is, by prescribing the rate of entropy dissipation ..."
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Abstract. We consider a class of nonconservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy. We formulate a wellposed Riemann problem by supplementing it with a kinetic relation, that is, by prescribing the rate of entropy dissipation across any shock wave. Our condition can be regarded as a generalization to nonconservative systems of a similar concept introduced by Truskinovsky and AbeyaratneKnowles for subsonic phase transitions and generalized by LeFloch for undercompressive waves to general hyperbolic systems. The proposed kinetic relation for nonconservative systems turns out to be equivalent, for the class of systems under consideration, to Dal Maso, LeFloch, and Murat’s definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase plane analysis of traveling solutions associated with an augmented version of the nonconservative system. We illustrate with several examples that nonconservative systems arising in the applications fit in our framework. For a typical model of turbulent fluid dynamics we provide a detailled analysis of the existence and properties of traveling waves and we derive the corresponding kinetic function. Numerical experiments illustrate the properties of the kinetic relations, which can serve to assess the efficiently of nonconservative schemes.
A godunovtype method for the shallow water equations with discontinuous topography in the resonant regime
 Journal of Computational Physics
, 2011
"... the resonant regime ..."
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The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous crosssection
 M2AN MATH. MODEL. NUMER. ANAL
, 2008
"... We consider the Euler equations for compressible fluids in a nozzle whose crosssection is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch, and Murat. Observing that th ..."
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We consider the Euler equations for compressible fluids in a nozzle whose crosssection is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch, and Murat. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class of numerical approximations for this system.
A FULLY WELLBALANCED, POSITIVE AND ENTROPYSATISFYING GODUNOVTYPE METHOD FOR THE SHALLOWWATER EQUATIONS
"... AMS subject classifications. 65M60, 65M12 Key words. Shallowwater equations, steady states, finite volume schemes, wellbalanced property, positive preserving scheme, entropy preserving scheme Abstract. This work is devoted to the derivation of a fully wellbalanced numerical scheme for the wellk ..."
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AMS subject classifications. 65M60, 65M12 Key words. Shallowwater equations, steady states, finite volume schemes, wellbalanced property, positive preserving scheme, entropy preserving scheme Abstract. This work is devoted to the derivation of a fully wellbalanced numerical scheme for the wellknown shallowwater model. During the last two decades, several wellbalanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the socalled lake at rest. By fully wellbalanced, we mean here that the proposed Godunovtype method is also able to preserve stationary states with non zero velocity. The numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality. 1. Introduction. The
THE MINIMUM ENTROPY PRINCIPLE FOR COMPRESSIBLE FLUID FLOWS IN A NOZZLE WITH DISCONTINUOUS CROSSSECTION
, 2007
"... Abstract. We consider the Euler equations for compressible fluids in a nozzle whose crosssection is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. ..."
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Abstract. We consider the Euler equations for compressible fluids in a nozzle whose crosssection is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. Pures Appl. 74 (1995) 483–548]. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class of numerical approximations for this system.
TESTING IMPROVEMENTS OF A WELLBALANCED METHOD FOR THE
"... Abstract. A set of improvements of a wellbalanced scheme for the model of fluid flows in a nozzle with variable crosssection is presented. Relying on the wellbalanced method introduced in our earlier work, we use the steady state solutions to absorb the nonconservative term. The underlying numeri ..."
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Abstract. A set of improvements of a wellbalanced scheme for the model of fluid flows in a nozzle with variable crosssection is presented. Relying on the wellbalanced method introduced in our earlier work, we use the steady state solutions to absorb the nonconservative term. The underlying numerical fluxes operating on these steady states are obtained as convex combinations of the numerical fluxes of a firstorder and a secondorder schemes. The improvements are still wellbalanced schemes. Then, we present many numerical tests, which establishes the efficiency of these numerical schemes. These schemes can provide us with very desirable approximations for any initial data: data in supersonic or subsonic regions, and data in both of these two kinds of regions. All the tests also show that the accuracy of the method by the improvements is improved. 1.