High Order SFV and Mixed SDG/FV Methods for the Uncertainty Quantification in Multidimensional Conservation Laws

Abstract

We analyze and apply the high order Stochastic Finite Volume (SFV) and mixed Stochastic Discontinuous Galerkin/Finite Volume (SDG/FV) methods used to quantify the uncertainty in hyperbolic conservation laws with random initial data and flux coefficients. We describe incomplete information in the conservation law mathematically as random fields. The SFVM is formulated to solve numerically the system of conservation laws with sources of randomness in both flux coefficients and initial data. We formulate the Stochastic Discontinuous Galerkin method which we primarily use to solve the multidimensional stochastic conservation laws on unstructured grids. Finally, we compare the efficiency of the SFV and SDG methods with of Monte-Carlo type methods. Finally, we introduce an adaptation technique based on the Karhunen-Loeve expansion of the random flux and/or initial data and apply it in order to reduce the computational cost of the SFVM.