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.

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.