A Study of Additive Linear Multistep Methods for Solving Advection-Diffusion-Reaction Equations

Abstract

This thesis investigates the efficacy of advanced numerical methods for solving advectiondiffusion- reaction (ADR) problems, essential for describing transport mechanisms in fluid or solid mediums. It encompasses a detailed study of variable stepsize, semi-implicit, backward differentiation formula (VSSBDF) methods up to fourth order and introduces novel 3-additive linear multistep methods, emphasizing the treatment of ADR models that are typically formulated as partial differential equations. By discretizing these into systems of ordinary differential equations for numerical analysis, this work addresses the challenges posed by stiff terms in ADR models. The first part of the thesis investigates the performance of VSSBDF methods of up to fourth order for solving ADR models employing two different implicit-explicit (IMEX) splitting approaches: a physics-based splitting and a splitting based on a dynamic linearization of the resulting system of ODEs , which is referred to as Jacobian splitting. 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 seven 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. The second part of the thesis gives a comprehensive analysis of 3-additive linear multistep methods, designed to separately treat diffusion, reaction, and advection components of differential equations that model problems in science and engineering. This approach addresses the limitations of conventional IMEX methods, which typically combine the three terms with only two numerical methods. The stability and order of convergence of these new methods are investigated, and their performance is compared with popular IMEX methods. The findings reveal that the new 3-additive methods generally offer larger stability regions and superior computational efficiency compared to the tested IMEX methods in certain cases. Together, these studies contribute to the field of numerical analysis by offering enhanced methods for the efficient and accurate solution of differential equations in science and engineering, reflecting significant advancements in the treatment of advection-diffusion-reaction systems.