Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data (Q2840000)

The authors consider scalar hyperbolic conservation laws in spatial dimension \( d\geq 1\) with stochastic initial data. Existence and uniqueness of a random-entropy solution are proved and some sufficient conditions on the initial data that ensure the existence of statistical moments of any order \( k\) of this random entropy solution are given. A class of numerical schemes of multi-level Monte Carlo finite volume (MLMC-FVM) type for the approximation of the ensemble average of the random entropy solutions as well as of their \( k\)-point space-time correlation functions are presented. 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. In certain cases, statistical moments of discontinuous solutions are found to be more regular than pathwise solutions.