## Preprint 2019-03

## Raimund Bürger, Daniel Inzunza, Pep Mulet, Luis M. Villada:

### Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour

### Abstract:

Nonlinear convection-diffusion equations with nonlocal flux and possibly degenerate diffusion describe interacting gases, flow in porous media, collective behaviour in biology, and other phenomena. Their numerical solution by an explicit finite difference method is costly since one needs to discretize a local spatial convolution for each evaluation of the convective numerical flux, and moreover the diffusion term gives rise to a disadvantageous Courant-Friedrichs-Lewy (CFL) condition. 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 [J.A. Carrillo, A. Chertock, Y. Huang, Commun. Comput. Phys. 17 (2015) 233--258]. The resulting IMEX-RK methods avoid the restrictive time step limitation of explicit schemes since the diffusion term is handled implicitly, but one needs to solve 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 Runge-Kutta schemes defining an IMEX-RK method. Results illustrate that for fine discretizations IMEX-RK methods are more efficient in terms of reduction of error versus CPU time than the original explicit method.

This preprint gave rise to the following definitive publication(s):

**Raimund BüRGER, Daniel INZUNZA, Pep MULET, Luis M. VILLADA: ***Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour*. Applied Numerical Mathematics, vol. 144, pp. 234-252, (2019).