A PROBABILISTIC SCHEME FOR SEMILINEAR NONLOCAL DIFFUSION EQUATIONS WITH VOLUME CONSTRAINTS

Minglei Yang, Guannan Zhang, Diego Del-Castillo-Negrete, Yanzhao Cao

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integrodifferential operator consists of both local convection-diffusion and nonlocal diffusion operators. Our numerical scheme is based on the direct approximation of the nonlinear Feynman-Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman-Kac representation avoids solving dense linear systems arising from nonlocal operators. Compared with existing stochastic approaches, our method can achieve first-order convergence after balancing the temporal and spatial discretization errors, which is a significant improvement of existing probabilistic/stochastic methods for nonlocal diffusion problems. Error analysis of our numerical scheme is established. The effectiveness of our approach is shown in two numerical examples. The first example considers a three-dimensional nonlocal diffusion equation to numerically verify the error analysis results. The second example presents a physics problem motivated by the study of heat transport in magnetically confined fusion plasmas.

Original languageEnglish
Pages (from-to)2718-2743
Number of pages26
JournalSIAM Journal on Numerical Analysis
Volume61
Issue number6
DOIs
StatePublished - 2023

Funding

\ast Received by the editors May 6, 2022; accepted for publication (in revised form) July 7, 2023; published electronically November 15, 2023. https://doi.org/10.1137/22M1494877 Funding: This work is partially supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, the Applied Mathematics Program under the grant DE-SC0022253 and the contract ERKJ387, and by Office of Fusion Energy Science via the Scientific Discovery through Advanced Computing (SciDAC) program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725.

Keywords

  • Brownian motion
  • Feynman-Kac formula
  • compound Poisson process
  • exit time
  • nonlocal diffusion equations
  • stochastic differential equation
  • transport

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