Numeric solution of advection–diffusion equations by a discrete time random walk scheme

• Published in: Numerical methods for partial differential equations Vol. 36; no. 3; pp. 680 - 704
• Main Authors: Angstmann, Christopher N., Henry, Bruce I., Jacobs, Byron A., McGann, Anna V.
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01-05-2020; Wiley Subscription Services, Inc
• Subjects: Advection; Advection-diffusion equation; Burgers' equation; Diffusion; discrete time random walk; Finite difference method; Nonlinear equations; numerical methods; Partial differential equations; Random walk; Stochastic models; Stochastic processes
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.22448

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 nonlinear 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 nonlinear 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.