Abstract

Stochastic conservation laws (SCL) with quasilinear multiplicative “rough” path dependence in the flux arise in modeling of mean field games. An impressive collection of theoretical results has been developed for SCL in recent years by Gess, Lions, Perthame, and Souganidis. We present the first fully computable numerical methods for pathwise solutions of scalar SCL with, for instance, “rough” paths in the form of Wiener processes. Convergence rates are derived for the numerical methods and we show that for strictly convex flux functions, “rough” path oscillations lead to cancellations in the solution flow map; a property we take advantage of to develop more efficient numerical methods.