Integrating-Factor-Based 2-Additive Runge–Kutta methods for Advection-Reaction-Diffusion Equations

There are three distinct processes that are predominant in models of flowing media with inter-acting components: advection, reaction, and diffusion. Collectively, these processes are typicallymodelled with partial differential equations (PDEs) known as advection-reaction-diffusion (ARD)equations.To solve most PDEs in practice, approximation methods known as numerical methods areused. The method of lines is used to approximate PDEs with systems of ordinary differentialequations (ODEs) by a process known as semi-discretization. ODEs are more readily analysedand benefit from well-developed numerical methods and software. Each term of an ODE thatcorresponds to one of the processes of an ARD equation benefits from particular mathematicalproperties in a numerical method. These properties are often mutually exclusive for many basicnumerical methods.A limitation to the widespread use of more complex numerical methods is that the developmentof the appropriate software to provide comparisons to existing numerical methods is not straight-forward. Scientific and numerical software is often inflexible, motivating the development of a classof software known as problem-solving environments (PSEs). Many existing PSEs such as Matlabhave solvers for ODEs and PDEs but lack specific features, beyond a scripting language, to readilyexperiment with novel or existing solution methods. The PSE developed during the course of thisthesis solves ODEs known as initial-value problems, where only the initial state is fully known.The PSE is used to assess the performance of new numerical methods for ODEs that integrateeach term of a semi-discretized ARD equation. This PSE is part of the PSE pythODE that usesobject-oriented and software-engineering techniques to allow implementations of many existing andnovel solution methods for ODEs with minimal effort spent on code modification and integration.The new numerical methods use a commutator-free exponential Runge–Kutta (CFERK) methodto solve the advection term of an ARD equation. A matrix exponential is used as the exponentialfunction, but CFERK methods can use other numerical methods that model the flowing medium.The reaction term is solved separately using an explicit Runge–Kutta method because solving italong with the diffusion term can result in stepsize restrictions and hence inefficiency. The diffusionterm is solved using a Runge–Kutta–Chebyshev method that takes advantage of the spatiallysymmetric nature of the diffusion process to avoid stepsize restrictions from a property knownas stiffness. The resulting methods, known as integrating-factor-based 2-additive-Runge–Kuttamethods, are shown to be able to find higher-accuracy solutions in less computational time thancompeting methods for certain challenging semi-discretized ARD equations. This demonstrates thepractical viability both of using CFERK methods for advection and a 3-splitting in general.