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We propose a novel numerical approach for nonlocal diffusion equations [8]with integrable kernels, based on the relationship between the backwardKolmogorov equation and backward stochastic differential equations (BSDEs)driven by L\`{e}vy processes with jumps. The nonlocal diffusion problem underconsideration is converted to a BSDE,for which numerical schemes are developedand applied directly. As a stochastic approach, the proposed method does notrequire the solution of linear systems, which allows for embarrassinglyparallel implementations and also enables adaptive approximation techniques tobe incorporated in a straightforward fashion. Moreover, our method is moreaccurate than classic stochastic approaches due to the use of high-ordertemporal and spatial discretization schemes. In addition, our approach canhandle a broad class of problems with general nonlinear forcing terms as longas they are globally Lipchitz continuous. Rigorous error analysis of the newmethod is provided as several numerical examples that illustrate theeffectiveness and efficiency of the proposed approach.

Numerical Methods for a Class of Nonlocal Diffusion Problems with the Use of Backward SDEs