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

Guannan Zhang, Weidong Zhao, Clayton Webster, Max Gunzburger

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8 Scopus citations

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.

Original languageEnglish
Pages (from-to)2479-2496
Number of pages18
JournalComputers and Mathematics with Applications
Volume71
Issue number11
DOIs
StatePublished - Jun 1 2016

Funding

This material is based upon work supported in part by the U.S. Air Force of Scientific Research under grant numbers FA9550-11-1-0149 and 1854-V521-12 ; by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract numbers ERKJ259, and ERKJE45; by the National Natural Science Foundation of China under grant numbers 91130003 , 11171189 and 11571206 ; by Natural Science Foundation of Shandong Province under grant number ZR2011AZ002 ; and by the Laboratory Directed Research and Development 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

  • Adaptive approximation
  • Backward stochastic differential equation with jumps
  • Compound Poisson process-scheme
  • Nonlocal diffusion equations
  • Superdiffusion

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