Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions are also presented.

中文翻译：

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守恒定律双曲系统的统计解：数值逼近

统计解是可积函数空间上的时间参数化概率测度，最近被提出作为多维双曲守恒定律系统的全局解和不确定性量化的框架。通过将高分辨率有限体积方法与蒙特卡罗采样程序相结合，我们提出了一种近似统计解的数值算法。在有限体积方法的可验证假设下，我们证明了由所提出算法生成的近似值在适当的拓扑中收敛到统计解。还介绍了说明收敛理论和揭示统计解决方案的有趣特性的数值实验。