### Schedule: July 29 - August 2 (2002) 2:00 pm - 4:30 pm

**Short description:**

This course will present numerical methods and theory for solving advection-diffusion-reaction
equations coupling advective and diffusive transport of chemically reacting
species. Such equations are found for example in environmental pollution
studies. The hyperbolic advection term in the equation then models transport
of one or more species in a velocity field in some medium, e.g. air
or water. The parabolic diffusion terms are mostly added as parameterizations
of turbulence and the reaction terms are ordinary differential equations
resulting from the mass-action law of chemical kinetics. Similar advection-diffusion-reaction
equations are found in biology and medicine. Here one often encounters
chemo-taxis models where gradient fields of bio-chemical species act
as velocity fields for populations under study. For example, existing
models for tumour angiogenesis and tumour invasion assume chemo-taxis.

The course will consist of 10 one-hour lectures, part of which are
introductory. The course is meant for graduate students, postdocs and
others unfamiliar with the advanced numerical solution of PDEs. Both
time stepping techniques and finite volume and finite element spatial
discretization methods will be discussed.

**Preliminary schedule:**

*Day 1* (Introductory, Jan Verwer):

Finite volume spatial discretization on Cartesian grids, including Fourier-von
Neumann stability analysis and limiting procedures for the advection
problem for getting positivity and monotonicity.

*Day 2* (Introductory, Martin Berzins):

Finite element discretizations. Introduction to continuous and discontinuous
Galerkin methods on regular and irregular grids. Application to advection-diffusion
reaction problems.

*Day 3* (More advanced, Jan Verwer):

Time integration methods based on operator splitting and alternatives,
such as implicit-explicit methods and Rosenbrock methods employing approximate
matrix factorization.

*Day 4* (More advanced, Martin Berzins):

Stabilized and nonlinear finite element methods for advection-diffusion
reaction problems: adjoint error estimation and continuous space-time
methods. Comparisons with finite volume methods.

*Day 5 *(More advanced, Jan Verwer):

Time integration continued. Amongst others stabilized Runge-Kutta methods
for diffusion-reaction equations.

An outline description of the course and references given by Martin
Berzin is available at:

the references
in pdf form

The two
lectures with four parts in pdf form are: Part 1 Part 2 Part 3 Part
4

###