A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems

Source abstract

Given that nonlocal diffusion (ND) problems allow their solutions to have spatial discontinuities, discontinuous Galerkin (DG) discretizations in space become natural choices when numerical approximations are considered. In this paper, we design and study a novel DG method for solving the one-dimensional time-dependent ND problem. The key idea of the method is the introduction of an auxiliary variable, analogous to the classic local discontinuous Galerkin (LDG) method but with some nonlocal extensions. Theoretical analysis shows that the proposed semi-discrete DG scheme is L 2 L^2 -stable, convergent, and in particular asymptotically compatible. This latter feature implies that the classical DG approximation of the local diffusion problem can be recovered in the zero horizon limit. We also present various numerical tests to demonstrate the effectiveness and robustness of our method.