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