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 27: Time discretizations for parabolic problems

Time dependent problems can be discretized with either explicit or implicit time stepping methods. In this lecture, I investigate the implications of using the simplest of these methods, the explicit forward Euler and the implicit Euler method, when applied to the prototypical parabolic equation: The heat equation. It turns out that explicit methods are completely impractical for parabolic problems because one has to choose the time step unreasonably small, and that one needs to use implicit methods.

The techniques presented in this lecture in fact apply for all parabolic equations and can be extended to other explicit or implicit time stepping methods, with similar results.

The same techniques will be used for hyperbolic equations in the next lecture, with completely opposite conclusions.

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Slides: click here