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Numerical solution of scalar conservation laws with random flux functions

by S. Mishra and N. H. Risebro and Ch. Schwab and S. Tokareva

(Report number 2012-35)

Abstract
We consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a Karhunen-Loève expansion on the state space of the random flux. For random flux functions which are Lipschitz continuous with respect to the state variable, we prove the existence of a unique random entropy solution. Using a Karhunen-Loève spectral decomposition of the random flux into principal components with respect to the state variables, we introduce a family of parametric, deterministic entropy solutions on high-dimensional parameter spaces. We prove bounds on the sensitivity of the parametric and of the random entropy solutions on the Karhunen-Loève parameters. We also outline the convergence analysis for two classes of discretization schemes, the Multi-Level Monte-Carlo Finite-Volume Method (MLMCFVM) developed in [22, 24, 23], and the stochastic collocation Finite Volume Method (SCFVM) of [25].

Keywords:

@Techreport{MRST12_478,
  author = {S. Mishra and N. H. Risebro and Ch. Schwab and S. Tokareva},
  title = {Numerical solution of scalar conservation laws with random flux functions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-35},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-35.pdf },
  year = {2012}
}

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