Numerical Solution of Scalar Conservation Laws with Random Flux Functions

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 continuously differentiable with respect to the state variable $u$, 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 multilevel Monte Carlo finite volume method (MLMCFVM), developed in [S. Mishra and C. Schwab, Math. Comp., 81 (2012), pp. 1979--2018], [S. Mishra, C. Schwab, and J. Šukys, J. Comput. Phys., 231 (2012), pp. 3365--3388], and [S. Mishra, C. Schwab, and J. Šukys, Multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws, in Uncertainty Quantification in Computational Fluid Dynamics, Lecture Notes in Comput. Sci. Eng. 92, Springer, Heidelberg, 2013, pp. 225--294], and the stochastic collocation finite volume method (SCFVM) of [S. Tokareva, Stochastic Finite Volume Methods for Computational Uncertainty Quantification in Hyperbolic Conservation Laws, Ph.D. dissertation, ETH Diss. Nr. 21498, ETH Zürich, Zürich, Switzerland]. (A corrected version is attached.)