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Numerical approximation of statistical solutions of scalar conservation laws

by U. S. Fjordholm and K. Lye and S. Mishra

(Report number 2017-52)

Abstract
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.

Keywords: Hyperbolic systems, statistical solutions, probability measures, Wasserstein Metric

@Techreport{FLM17_748,
  author = {U. S. Fjordholm and K. Lye and S. Mishra},
  title = {Numerical approximation of statistical solutions of scalar conservation laws},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-52},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-52.pdf },
  year = {2017}
}

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