Asymptotically compatible schemes for space-time nonlocal diffusion equations

In this paper, we study a space-time nonlocal diffusion model that reduces to the classical diffusion equation in the local limit. Firstly, we show the uniqueness and existence of the weak solution of the nonlocal model, and study the local limit of the nonlocal model as horizon parameters approach zero. Particularly, it is shown that the convergence is uniform at a rate of O (delta + sigma(2)), under certain regularity assumptions on initial and source data. Next we propose a fully discrete scheme, by exploiting the quadrature-based finite difference method in time and the Fourier spectral method in space, and show its stability. The numerical scheme is proved to be asymptotically compatible so that it preserves the local limiting behavior at the discrete level. Numerical experiments are provided to illustrate the theoretical results. (C) 2017 Elsevier Ltd. All rights reserved.