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Statistical solutions of hyperbolic systems of conservation laws: Numerical approximation

U. S. Fjordholm

Department of Mathematics, University of Oslo, Postboks 1053 Blindern, 0316 Oslo, Norway

E-mail Address: ulriksf@math.uio.no

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K. Lye

Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

E-mail Address: kjetil.lye@sam.math.ethz.ch

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S. Mishra

Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

E-mail Address: siddhartha.mishra@sam.math.ethz.ch

Corresponding author.

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, and

F. Weber

Department of Mathematical Sciences, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA

E-mail Address: franzisw@andrew.cmu.edu

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https://doi.org/10.1142/S0218202520500141Cited by:20 (Source: Crossref)

Abstract

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.

Communicated by F. Brezzi

Keywords:

AMSC: 65M08, 65C05, 65C30, 35L65, 76N10, 76F65

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