Analysis of a line method for reaction-diffusion models of nonlocal type

The considerable interest in the recent literature for nonlocal equations and their applications to a wide range of scientific problems does not appear to be supported by a corresponding advancement in the efforts toward reliable numerical techniques for their solution. The aim of this paper is to provide such an algorithm and above all to prove its high order convergence. The numerical scheme is applied to a diffusion problem in a biological context, arising in population theory. Several simulations are carried out. At first the empirical order of convergence on examples with known solution is assessed. Their results are in agreement with the theoretical findings. Simulations are then extended to cases for which the solution is not known a priori. Also in this case the outcomes support the convergence analysis.