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On the convergence of numerical schemes for hyperbolic systems of conservation laws

We survey convergence results for numerical methods approximating hyperbolic systems of conservation laws, a large class of nonlinear PDEs that arise in physics and engineering. We introduce a novel solution concept, that of statistical solutions, which are more appropriate as a framework for solutions and uncertainty quantificatio for multi-dimensional systems of conservation laws. An ensemble averaging Monte Carlo type algorithm is proposed and proved to converge to statistical solutions on mesh refinement.

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