Sparse Tensor Multi-Level Monte Carlo Finite Volume Methods for hyperbolic conservation laws with random initial data

Abstract

We consider scalar hyperbolic conservation laws in several $(d \ge 1)$ spatial dimensions with stochastic initial data. We prove existence and uniqueness of a random-entropy solution and show existence of statistical moments of any order $k$ of this random entropy solution. We present a class of numerical schemes of multi-level Monte Carlo Finite Volume (MLMC-FVM) type for the approximation of random entropy solutions as well as of their $k$-point correlation functions. These schemes are shown to obey the same accuracy vs. work estimate as a single application of the finite volume solver for the corresponding deterministic problem. Numerical experiments demonstrating the efficiency of these schemes are presented. Statistical moments of discontinuous solutions are found to be more regular than pathwise solutions.