Numerical method for hyperbolic conservation laws via forward backward SDEs

TL;DR

This paper introduces a novel numerical method for hyperbolic conservation laws that leverages probabilistic interpretations via forward-backward stochastic differential equations, enabling high accuracy solutions with parallel computation and relaxed stability conditions.

Contribution

The work presents a new probabilistic approach using Feynman-Kac formulas and forward-backward SDEs to efficiently compute viscosity solutions of hyperbolic conservation laws.

Findings

Method allows small viscosity parameter (e.g., 10^{-10})

Computations are fully parallelizable at each time step

Traditional CFL condition is significantly weakened

Abstract

It is well known that for solutions of semi-linear parabolic PDEs, there are equivalent probabilistic interpretations, which yields the so called nonlinear Feymman-Kac formula. By adopting such formula, we consider in this work a novel numerical approach for solutions of hyperbolic conservation laws. Our numerical method consists in efficiently computing the viscosity solutions of conservation laws. However, instead of solving the viscosity problem directly (which is difficult), we find its equivalent probabilistic solution by adopting the Feymman-Kac formula, which relies on solving the equivalent forward backward stochastic differential equations. It is noticed that such framework possesses the following advantages: (i) the viscosity parameter can be chosen sufficiently small (say $10^{-10}$); (ii) the computational procedure on each discretized time level can be \textit{completely parallel}; (iii) the traditional CFL condition is dramatically weakened; (iv) one does not need to handle the transition layers and discertizations of derivatives. Thus, high accuracy viscosity solutions can be efficiently found. Several numerical examples are given to demonstrate the effectiveness of the proposed numerical method.