Mathematical Sciences: Hyperbolic Systems of Conservation Laws: Theory and Applications

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9401003 Le Floch This proposal is motivated by the numerical difficulties arising in Computational Fluid Dynamics, and more generally in continuum mechanics, when computing solutions to nonlinear hyperbolic systems of PDE's. The solutions to such a system are discontinuous, and, being not uniquely determined by their initial data, are selected via the so-called entropy criterion. While the entropy criterion is treated in the literature as an extra condition that is checked afterwards, the P.I. developed approaches where the entropy criterion plays the very central role in analyzing the properties of a scheme, or in the derivation of the scheme itself. The emphasis here will be on establishing the entropy criterion and the nonlinear stability of high order accurate schemes for multidimensional systems of conservation laws. This work is aimed at clarifying the role of the so-called entropy criterion for the numerical approximation of hyperbolic systems of conservation laws. The P.I. developed new methods of analysis and numerical analysis, and established that several numerical schemes are consistent with the entropy criterion. Such a result guarantees that a method is nonlinearly stable, and produces the physically meaningful discontinuous solutions. The techniques also provide new insight on the schemes used for shock wave calculations, and suggest possible improvements. The focus here will be on developing flux-splitting methods, second-order Godunov-type schemes, and truly multidimensional schemes for systems of conservation laws. The analysis will include concept such as: entropy dissipation estimates, tracking of monotonicity regions, the compensated compactness method, the measure-valued solutions, etc. Further insight about the nonlinear stability will be obtained by considering the issue of uniqueness and continuous dependence with respect to the initial condition.