Gradient-based Monte Carlo methods for relaxation approximations of hyperbolic conservation laws

TL;DR

This paper introduces gradient-based Monte Carlo methods for hyperbolic conservation laws, extending particle derivative evolution techniques to improve variance reduction and shock structure representation in numerical simulations.

Contribution

It generalizes gradient random walk methods to hyperbolic systems using relaxation approximations and asymptotic-preserving discretization, offering a novel particle dynamics approach.

Findings

Significant variance reduction compared to standard methods

Enhanced shock structure resolution in simulations

Effective in one-dimensional scalar and system cases

Abstract

Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier--Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the '90s and have several interesting features, such as being grid free, automatically adapting to the solution by concentrating elements where the gradient is large and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws. To achieve this goal, we first extend the classical Monte Carlo method to relaxation approximation of systems of conservation laws, and subsequently consider a novel particle dynamics based on the spatial derivatives of the solution. The methodology, combined with asymptotic-preserving splitting discretization, yields a way to construct a new class of gradient-based Monte Carlo methods for hyperbolic systems of conservation laws. Several results in one spatial dimension for scalar equations and systems of conservation laws show that the new methods are very promising and yield remarkable improvements compared to standard Monte Carlo approaches, either in terms of variance reduction as well as in describing the shock structure.