Performance comparison of variable-stepsize IMEX SBDF methods on advection-diffusion-reaction models

Advection-diffusion-reaction (ADR) models describe transport mechanisms in fluid or solid media. They are often formulated as partial differential equations that are spatially discretized into systems of ordinary differential equations (ODEs) in time for numerical resolution. This paper investigates the performance of variable stepsize, semi-implicit, backward differentiation formula (VSSBDF) methods of up to fourth order for solving ADR models employing two different implicit-explicit splitting approaches: a physics-based splitting and a splitting based on a dynamic linearization of the resulting system of ODEs, called jacobian splitting in this paper. We develop an adaptive time-stepping and error control algorithm for VSSBDF methods up to fourth order based on a step-doubling refinement technique using estimates of the local truncation errors. Through a systematic comparison between physics-based and Jacobian splitting across six ADR test models, we evaluate the performance based on CPU times and corresponding accuracy. Our findings demonstrate the general superiority of Jacobian splitting in several experiments.