Scope: The course will provide a solid foundation on
traditional and advanced numerical discretisation techniques
for the solution of the various partial differential
equations which govern fluid flow, showing how successful
numerical methods are rooted both in mathematical properties
of those equations and in notions of computational efficiency.

Topics to be covered include:

    * Brief history of CFD.
    * Discrete representation of continua.
    * Properties of an ideal CFD code.
    * Consistency and stability.
    * The one-dimensional heat equation.
    * Analytical properties.
    * Explicit and implicit schemes.
    * Tridiagonal matrices.
    * Accuracy and stability.
    * Classification of partial differential equations.
Marching and jury problems.
    * Fourier analysis and discrete Fourier analysis.
    * Laplace's equation. Analytical properties. Numerical
schemes. Convergence to the steady state and the most
persistent error mode.
    * A guide to multigrid.
    * The linear advection problem.
    * Phase and dissipation errors.
    * Upwinding.
    * Q-schemes.
    * Advection-diffusion and boundary layers.
    * Godunov's Theorem and how to evade it.
    * Flux Corrected Transport and limiters.
    * Linear systems of equations.
    * Conservation laws.
    * The traffic equation.
    * Shock waves, conservation form and finite-volume schemes.
    * Enforcing entropy conditions.
    * Systems of conservation laws.
    * Characteristics and characteristic equations. The
Riemann problem. Godunov-type schemes.
    * Numerical fluxes.
    * Kinetic schemes.
    * Robustness. Multidimensional problems.
    * Structured, unstructured, and moving grids.
    * Handling geometry.

Working Plan: The morning lectures on the theory will be
complemented by hands-on computational practicals during the
afternoons. The participants will work on numerical
exercises as individuals and as members of a team, and
present their results to the rest of the class.

There will also be seminars by established researchers on
application of CFD to contemporary science and technology
topics.

Who should attend: The course will aim to convey a "CFD
literacy" enabling newcomers to understand the literature
and to select methods that are effective for their
applications. It will also provide the foundation for
existing users who would like to become more involved with
algorithm and code development. A strong background in an
applied science or/and mathematics is recommended.

At the end of the course the participants will be qualified
to start building their own code from scratch, or to further
develop existing packages.