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 ran- dom (and parametric) flux functions. We propose a Karhunen-Loeve 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-Loeve 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-Loeve 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. ˇ Sukys, J. Comput. Phys., 231 (2012), pp. 3365-3388), and (S. Mishra, C. Schwab, and J. ˇ Multi-level Monte Carlo finite volume methods for uncertainty quantifi- cation 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. dis- sertation, ETH Diss. Nr. 21498, ETH Zurich, Zurich, Switzerland).