Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour.
- Resource Type
- Article
- Authors
- Bürger, Raimund; Inzunza, Daniel; Mulet, Pep; Villada, Luis M.
- Source
- Applied Numerical Mathematics. Oct2019, Vol. 144, p234-252. 19p.
- Subject
- *FINITE differences
*TRANSPORT equation
*FINITE difference method
*NONLINEAR difference equations
*NONLINEAR equations
*EQUATIONS
*NONLINEAR systems
- Language
- ISSN
- 0168-9274
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 Runge-Kutta schemes defining an IMEX-RK method. For fine discretizations, IMEX-RK methods turn out more efficient in terms of reduction of error versus CPU time than the original explicit method. [ABSTRACT FROM AUTHOR]