Efficiency of 3-Additive Linear Multistep Methods for a Stiff Combustion Model

Efficient and accurate time integration is essential for simulating combustion processes with stiff, multiscale dynamics arising from diffusion, reaction, and advection interactions. Standard implicit–explicit (IMEX) methods employ a two-additive splitting in which these three physical processes are grouped into only two components, often forcing diffusion and reaction to be treated together despite their different numerical characteristics. This coupling may limit computational efficiency and flexibility. Three-additive splitting methods overcome this limitation by treating diffusion, reaction, and advection as separate components, allowing each process to be integrated using a more suitable implicit or explicit method. Although the theoretical properties of three-additive linear multistep methods (LMMs) have previously been established, their performance has been assessed only on a single benchmark problem. This paper presents the first systematic numerical evaluation of implicit–implicit–explicit (IIE) and implicit–explicit–explicit (IEE) linear multistep methods for a stiff one-dimensional combustion model with FKPP, Ignition, and Fisher reaction kinetics. The three-additive splitting methods are compared with standard IMEX-LMMs of up to fourth order using CPU time, RMS error, and a model-based stability analysis. The results show that the IIE and IEE methods consistently improve computational efficiency while maintaining comparable numerical accuracy and stability. In particular, the first-order methods exhibit larger stability regions than their IMEX counterparts, whereas the higher-order methods retain comparable stability while achieving superior work–precision performance. These findings demonstrate that three-additive LMMs provide an efficient and flexible alternative to conventional IMEX methods for stiff diffusion–reaction–advection systems.