Starting in the spring 2013, I videotaped the lectures for my MATH 676: Finite element methods in scientific computing course at the KAMU TV studio at Texas A&M. These are lectures on many aspects of scientific computing, software, and the practical aspects of the finite element method, as well as their implementation in the deal.II software library. Support for creating these videos was also provided by the National Science Foundation and the Computational Infrastructure in Geodynamics.

The videos are part of a broader effort to develop a modern way of teaching Computational Science and Engineering (CS&E) courses. If you are interested in adapting our approach, you may be interested in this paper I wrote with a number of education researchers about the structure of such courses and how they work.

Note 1: In some of the videos, I demonstrate code or user interfaces. If you can't read the text, change the video quality by clicking on the "gear" symbol at the bottom right of the YouTube player.

Note 2: deal.II is an actively developed library, and in the course of this development we occasionally deprecate and remove functionality. In some cases, this implies that we also change tutorial programs, but the nature of videos is that this is not reflected in something that may have been recorded years ago. If in doubt, consult the current version of the tutorial.

Lecture 30.25: Time discretizations for advection-diffusion and other problems: IMEX, operator splitting, and other ideas

Having looked at (explicit and implicit) time discretizations for time dependent model problems in the last few lectures, I turn to the question of what to do in slightly more complicated cases where the equation is not either clearly hyperbolic or parabolic. In fact, in reality many equations lie somewhere on a spectrum, depending on whether transport or diffusion are the dominant factors, and we may have to treat the respective terms differently. In other cases, equations are coupled or simply have a completely different structure.
This lecture then is about what to do in such situations, and introduces a few ideas that explain how one approaches such cases. In particular, I introduce IMEX (implicit-explicit) and operater splitting schemes (e.g., the "Lie" and "Strang" splitting methods). I then also discuss a number of ways how one can increase the accuracy of these methods without too much additional effort.