Numerical approximation of statistical solutions of scalar conservation laws

30 Oct 2017  ·  Ulrik Skre Fjordholm, Kjetil Lye, Siddhartha Mishra ·

We propose efficient numerical algorithms for approximating statistical solutions of scalar conservation laws. The proposed algorithms combine finite volume spatio-temporal approximations with Monte Carlo and multi-level Monte Carlo discretizations of the probability space. Both sets of methods are proved to converge to the entropy statistical solution. We also prove that there is a considerable gain in efficiency resulting from the multi-level Monte Carlo method over the standard Monte Carlo method. Numerical experiments illustrating the ability of both methods to accurately compute multi-point statistical quantities of interest are also presented.

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Numerical Analysis 35L65, 65M08, 65C05, 65C30