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hyperbolic models and numerical analysis for shallow water flows

Emmanuel Audusse 1
1 BANG - Nonlinear Analysis for Biology and Geophysical flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt
Abstract : In this work we study some hyperbolic conservation laws related to shallow water flows.
First we consider the Saint-Venant system with source terms and we develop a second order bidimensional well-balanced finite volumes scheme that is based on a kinetic interpretation of the system and on a hydrostatic reconstruction of the interfaces values. The scheme is consistent and conservative and it preserves the non-negativity of the water height.
Then we extend the kinetic interpretation to the coupling with a transport equation. We construct a two time steps scheme that takes into account all the eigenvalues of the problem. This approach preserves the stability properties of the system and reduces the numerical diffusion and the computational cost.\\
We also present a new multilayer Saint-Venant system that allows us to obtain non constant vertical velocity profiles while preserving an invariant two dimensional domain of calculation. We present the derivation of the system and we study its stability - energy, hyperbolicity. We also investigate its relation with other fluid models and we perform its numerical implementation.
Finally we prove a uniqueness theorem for scalar conservation laws with discontinuous flux. Our proof uses a new family of entropies that are a natural way to adapt classical Kruzkov's entropies to the discontinuous case. This new method avoids the making of some classical hypothesis on the flux such as convexity, BV bounds or finite number of dicontinuities and does not need the definition of some interface condistion.
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Submitted on : Thursday, January 13, 2005 - 2:47:04 PM
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Emmanuel Audusse. hyperbolic models and numerical analysis for shallow water flows. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2004. English. ⟨tel-00008047⟩

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