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Dispersive shock waves and modulation theory

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock-wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, e.g., traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation, hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical one-dimensional (1D) dissipative and dispersive shock problem shows significant differences in shock structure and shock-front speed. Numerical results associated with the three-dimensional experiment show that three- and two-dimensional approximations are in excellent agreement and 1D approximations are in good qualitative agreement. Using 1D DSW theory, it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.

Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics

We present a model for the o w of pedestrians that describes features typical of this o w, such as the fall due to panic in the outo w of people through a door. The mathematical techniques essentially depend on the use of nonclassical shocks in scalar conservation laws. 2000 Mathematics Subject Classic ation: 35L65, 90B20.

Pedestrian flows and non-classical shocks

We propose a way to construct robust numerical schemes for the computations of numerical solutions of one- and two-dimensional hyperbolic systems of balance laws. In order to reduce the computational cost, we selected the family of flux vector splitting schemes. We reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations. This is applied to a 1D shallow water system with topography and to a 2D simplified model of two-phase flows with damping effects. Numerical results and comparisons with a classical centered discretizations scheme are supplied.

A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms☆

We deal with a single conservation law in one space dimension whose flux function is discontinuous in the space variable and we introduce a proper framework of entropy solutions. We consider a large class of fluxes, namely, fluxes of the convex-convex type and of the concave-convex (mixed) type. The alternative entropy framework that is proposed here is based on a two step approach. In the first step, infinitely many classes of entropy solutions are defined, each associated with an interface connection. We show that each of these class of entropy solutions form a contractive semigroup in L1 and is hence unique. Godunov type schemes based on solutions of the Riemann problem are designed and shown to converge to each class of these entropy solutions. The second step is to choose one of these classes of solutions. This choice depends on the Physics of the problem being considered and we concentrate on the model of two-phase flows in a heterogeneous porous medium. We define an optimization problem on the set of admissible interface connections and show the existence of an unique optimal connection and its corresponding optimal entropy solution. The optimal entropy solution is consistent with the expected solutions for two-phase flows in heterogeneous porous media.

Philippe G. LeFloch

Papers

Hyperbolic Systems of Conservation Laws