A Probabilistic Scheme for Semilinear Nonlocal Diffusion Equations with Volume Constraints

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

doi:10.1137/22m1494877

Title: A Probabilistic Scheme for Semilinear Nonlocal Diffusion Equations with Volume Constraints

Journal Article · Wed Nov 15 00:00:00 EST 2023 · SIAM Journal on Numerical Analysis

DOI:https://doi.org/10.1137/22m1494877· OSTI ID:2251656

Yang, Minglei ^[1];

^[1]; Del-Castillo-Negrete, Diego ^[1]; Cao, Yanzhao ^[2]

Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Auburn Univ., AL (United States)

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 integro-differential operator consists of both local convection-diffusion and nonlocal diffusion operators. Here, 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.

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Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC)

Grant/Contract Number:: AC05-00OR22725; SC0022253

OSTI ID:: 2251656

Journal Information:: SIAM Journal on Numerical Analysis, Vol. 61, Issue 6; ISSN 0036-1429

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

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