Lecture Notes
On Discontinuous Galerkin
methods
for convection-dominated problems
These methods have been devised to solve
nonlinear
hyperbolic systems and convection-dominated problems.
Examples
of problems to which these methods can be applied are:
- the Euler equations of gas dynamics,
- the shallow water equations,
- the equations of magneto-hydrodynamics,
- the compressible Navier-Stokes equations with high Reynolds
numbers,
- the equations of the hydrodynamic model for semiconductor device
simulation.
The main features that make the methods
attractive are:
- their formal high-order accuracy,
- their nonlinear stability,
- their high parallelizability,
- their ability to handle complicated geometries, and
- their ability to capture the discontinuities or strong gradients
of the
exact solution without producing spurious oscillations.
The gzipped postscript file
containing my 1997 C.I.M.E. lecture notes (# 1697) (7.2 MB) can
be downloaded by clicking here.
The gzipped postscript file
containing my 1999 NATO/von Karman Institute for Fluid Dynamics/ NASA
(LNCSE volume 9) lecture notes (8.7 MB) can be downloaded by ftp
by
clicking here.
Comments on the notes are very much appreciated;
please send them by e-mail to cockburn@math.umn.edu.
A simple introduction to error
estimation
for nonlinear hyperbolic conservation laws
In these notes, we present a simple
introduction
to the topic of a posteriorierror
estimation for nonlinear hyperbolic conservation laws. This
is a topic of great practical interest which has been receiving
increasing
attention from many researchers in the last several years. On the other
hand, the highly complex character of its mathematics often obscures
the
main ideas behind the technical manipulations. Aware of this
unfortunate
situation, we have written these notes in an attempt to emphasize
the ideas and
simplify the presentation
of the techniques. The reader has to be warned, however,
that these are notes on ongoing research-
not about fully solved problems.
The gzipped postscript file
containing the lecture notes (176 KB) can be retrieved by ftp by
clicking here;
it has 48 pages with one figure. Comments on the notes are very much
appreciated;
please send them by e-mail to cockburn@math.umn.edu.