Numeric Solution of Advection-Diffusion Equations by a Discrete Time Random Walk Scheme

TL;DR

This paper introduces a stable, easy-to-implement explicit numerical scheme for solving non-linear advection-diffusion equations with shocks, based on a discrete time and space stochastic process.

Contribution

It presents a novel stochastic process-based scheme that guarantees stability and accuracy for equations with shocks, overcoming limitations of traditional finite difference methods.

Findings

The scheme is stable for non-linear advection-diffusion equations.

Examples demonstrate the scheme's effectiveness in handling shocks.

The stochastic process formulation is crucial for stability and accuracy.

Abstract

Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups and discontinuities. Here we present an explicit numerical scheme for solving non-linear advection-diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for non-linear advection-diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme.