A powerful method for the numerical soluion of partial differential equations (PDEs) is the method of lines. Typically, when using the method of lines, the spatial domain is discretized on a grid using, e.g., finite differences, finite elements, or spectral methods, and a large structured set of coupled ordinary differential equations (ODEs) is obtained for the evolution of the solution at the grid points. These ODEs can then be evolved in time by your favourite numerical integration method. A major advantage provided by the method of lines is this flexibility in choosing the integration method. For example one can take advantage of the many high quality ODE software packages available and all the concomitant features such as stepsize and error control, continuous interpolants, and event location. However almost without exception, these software packages are based on classical integration methods; i.e., they are either fully explicit or fully implicit methods. ODEs derived from a method-of-lines discretization of PDEs often contain terms of differing nature based on their physical origins; e.g., convection and diffusion. In such cases, it is natural to integrate these terms by different numerical methods. In particular, diffusion is often stiff and linear, whereas convection is often non-stiff and non-linear. This is an ideal scenario for the use of an implicit-explicit (IMEX) method: the diffusion terms can be treated implicitly, and the convection terms can be treated explicitly. IMEX methods can typically take a much larger stable time-step than a fully explicit method, and in the case of linear diffusion, the avoid the Newton iterations (and the added costs and potential problems associated with them) of a fully implicit method. In this lecture, we motivate the use of IMEX methods for the numerical integration of ODEs, in particular those arising from a method-of-lines discretization of PDEs. We begin by establishing notation, conventions, and fundamental concepts to be used in this and the two subsequent lectures on IMEX methods. We touch on related concepts such as numerical stability and stiffness. We conclude this lecture by giving a few head-to-head comparisons between IMEX and non-IMEX methods.
In this lecture, we continue our treatment of IMEX methods by describing many different methods that have been proposed in the literature that can be classified as IMEX methods. Some of these methods include partitioned or additive Runge-Kutta methods, IMEX multi-step methods, splitting (fractional step) methods, integrating factor methods, the so-called ``sliders'' method, exponential time differencing, and spectral deferred correction. We attempt to describe the particular advantages and weaknesses of the various methods, in particular with respect to the classes of problems for which they are best suited.
In this lecture, we conclude our treatment of IMEX methods by discussing the many implementational aspects of numerical integration methods, with special attention paid to the relevant details for the IMEX methods described in the previous lectures. These details include for example the design and/or choice of methods, error control, and linear and/or nonlinear equations solvers. Time permitting, some discussion of implementational aspects in parallel or distributed computing environments will be included.