Mathematical Modelling

Lecture content:

This lecture introduces the use of mathematical models as a tool of studying natural system. Its aim is to present the necessary mathematical techniques to be able to understand/manipulate and change existing models in the environmental sciences.

The course is organized along classical and relatively simple models from selected fields in the earth sciences (oceanography, biogeochemistry, ecology). These models are used as examples to explain the predominantly numerical mathematics that are needed for solving them, and that are also used in more complex models. Every few lectures we will have a practical exercise where students write / change models that are given as small computer programs in MATLAB or OCTAVE and discuss the results.
The lecture consists of the following five blocks:

I Introduction: What is a Model?

II Static models
- Equilibrium distributions and box models
- Balance equations and steady states

III A dynamic view of the world
- Time-discrete models
- Dynamical models with one/several variables
- Numerical solution of ordinary differential equations
- Boundary value problems: shooting vs. relaxation

IV Data and models
- Model fitting: How to measure model-data disagreement
- Bayesian approach to parameter estimation
- Optimization methods and model uncertainties
V Models in time and space

- Classification of partial differential equations (PDEs)
- Solving PDE's with finite differences
- Solving PDE's with spectral methods
- A first look into large-scale circulation models
(Special case: shallow water equations)

The course is self-contained, so there is no mandatory reading. The treatment of numerical methods is based strongly on the book 'Numerical Recipes' by William H. Press, Saul Teukolsky, William T. Vetterling und Brian P. Flannery