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Statistical solutions of hyperbolic systems of conservation laws: numerical approximation

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

(Report number 2019-28)

Abstract
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that 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.

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

@Techreport{FLMW19_832,
  author = {U. S. Fjordholm and K. Lye and S. Mishra and F. Weber},
  title = {Statistical solutions of hyperbolic systems of conservation laws: numerical approximation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-28.pdf },
  year = {2019}
}

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