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.
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 31: First-order hyperbolic systems
First-order hyperbolic systems are peculiar in many regards, and rather different from the usual elliptic, parabolic and second-order hyperbolic problems we have considered so far. When using a naive discretization, one realizes quickly that the numerical solution is oscillatory and that the oscillations grow rather than decay as the mesh is refined. We look at the underlying causes (which are already present, in parts, in the continuous problem) and common approaches to mitigating them with numerical methods appropriate to first-order hyperbolic problems.
Due to time pressure, I managed to forget to discuss one of the most important parts of the lecture: Godunov's theorem, as shown on the very last slide of the PDFs for this lecture, linked to below.
Slides: click here