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

Christopher N. Angstmann, Bruce I. Henry, Byron A. Jacobs, Anna V. McGann

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)680-704
Number of pages25
JournalNumerical Methods for Partial Differential Equations
Issue number3
Publication statusPublished - 1 May 2020
Externally publishedYes


  • Burgers' equation
  • discrete time random walk
  • numerical methods

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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