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Numeric solution of advection–diffusion equations by a discrete time random walk scheme
Author(s) -
Angstmann Christopher N.,
Henry Bruce I.,
Jacobs Byron A.,
McGann Anna V.
Publication year - 2020
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22448
Subject(s) - mathematics , nonlinear system , advection , classification of discontinuities , diffusion process , stability (learning theory) , random walk , partial differential equation , diffusion , shock (circulatory) , numerical stability , stochastic differential equation , mathematical analysis , numerical analysis , computer science , innovation diffusion , medicine , knowledge management , physics , quantum mechanics , machine learning , thermodynamics , statistics
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.