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Hyperbolic Conservation Laws with Stochastic Discontinuous Flux Functions

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Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples (FVCA 2020)

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Abstract

Hyperbolic conservation laws are utilized to describe a variety of real-world applications, which require the consideration of the influence of uncertain parameters on the solution to the problem. To extend these models, one is often interested in including discontinuities in the state space to the flux function of the conservation law. This paper studies the solution of a stochastic nonlinear hyperbolic partial differential equation (PDE), whose flux function contains random spatial discontinuities. The first part of the paper defines the corresponding stochastic adapted entropy solution and required properties for existence and uniqueness are addressed. The second part contains the numerical simulation of the nonlinear hyperbolic problem as well as the estimation of the expectation of the problem via the multilevel Monte Carlo method.

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Acknowledgements

Funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC 2075—390740016.

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Authors and Affiliations

  1. University of Stuttgart, Allmandring 5b, 70569, Stuttgart, Germany

    Lukas Brencher & Andrea Barth

Authors
  1. Lukas Brencher
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  2. Andrea Barth
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Correspondence to Lukas Brencher .

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Editors and Affiliations

  1. NORCE Norwegian Research Centre AS, Bergen, Norway

    Robert Klöfkorn

  2. Department of Mathematics, University of Bergen, Bergen, Norway

    Eirik Keilegavlen

  3. Department of Mathematics, University of Bergen, Bergen, Norway

    Florin A. Radu

  4. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Jürgen Fuhrmann

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Brencher, L., Barth, A. (2020). Hyperbolic Conservation Laws with Stochastic Discontinuous Flux Functions. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_23

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