3-additive linear multi-step methods for diffusion-reaction-advection models

Abstract

Some systems of differential equations that model problems in science and engineering have natural splittings of the right-hand side into the sum of three parts, in particular, diffusion, reaction, and advection. Implicit-explicit (IMEX) methods treat these three terms with only two numerical methods, and this may not be desirable. Accordingly, this work gives a detailed study of 3-additive linear multi-step methods for the solution of diffusion-reaction-advection systems. Specifically, we construct new 3-additive linear multi-step methods that treat diffusion, reaction, and advection with separate methods. The stability of the new methods is investigated, and the order of convergence is tested numerically. A comparison of the new methods is made with some popular IMEX methods in terms of stability and performance. It is found that the new 3-additive methods have larger stability regions than the IMEX methods tested in some cases and generally outperform in terms of computational efficiency.