Numerical methods for a class of nonlocal diffusion problems with the use of backward SDEs

Abstract

We propose a novel numerical approach for nonlocal diffusion equations Du et al. (2012) with integrable kernels, based on the relationship between the backward Kolmogorov equation and backward stochastic differential equations (BSDEs) driven by Lèvy jumps processes. The nonlocal diffusion problem under consideration is converted to a BSDE, for which numerical schemes are developed. As a stochastic approach, the proposed method completely avoids the challenge of iteratively solving non-sparse linear systems, arising from the nature of nonlocality. This allows for embarrassingly parallel implementation and also enables adaptive approximation techniques to be incorporated in a straightforward fashion. Moreover, our method recovers the convergence rates of classic deterministic approaches (e.g. finite element or collocation methods), due to the use of high-order temporal and spatial discretization schemes. In addition, our approach can handle a broad class of problems with general inhomogeneous forcing terms as long as they are globally Lipschitz continuous. Rigorous error analysis of the new method is provided as several numerical examples that illustrate the effectiveness and efficiency of the proposed approach.