Abstract

The numerical solution of nonlinear convection-diffusion equations with nonlocal flux by explicit finite difference methods is costly due to the local spatial convolution within the convective numerical flux and the disadvantageous Courant-Friedrichs-Lewy (CFL) condition caused by the diffusion term. More efficient numerical methods are obtained by applying second-order implicit-explicit (IMEX) Runge-Kutta time discretizations to an available explicit scheme for such models in Carrillo et al. (2015) [13]. The resulting IMEX-RK methods require solving nonlinear algebraic systems in every time step. It is proven, for a general number of space dimensions, that this method is well defined. Numerical experiments for spatially two-dimensional problems motivated by models of collective behaviour are conducted with several alternative choices of the pair of RungeKutta schemes defining an IMEX-RK method. For fine discretizations, 1MEX-RK methods turn out more efficient in terms of reduction of error versus CPU time than the original explicit method. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.