A probabilistic scheme for semilinear nonlocal diffusion equations with volume constraints

TL;DR

This paper introduces a probabilistic numerical scheme for solving semilinear nonlocal diffusion equations with volume constraints, leveraging the Feynman-Kac formula to improve convergence and computational efficiency.

Contribution

The work develops a novel probabilistic approach based on the nonlinear Feynman-Kac formula that avoids dense linear systems and achieves first-order convergence for nonlocal diffusion problems.

Findings

Achieves first-order convergence in numerical solutions.

Successfully applies to 3D nonlocal diffusion and heat transport problems.

Error analysis confirms the method's accuracy and efficiency.

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 integro-differential 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 successfully avoids solving dense linear systems arising from nonlocality 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.