(Created 2010-07-25.)
 FINITE VOLUME METHODS FMN091
Credits: 7,5. Grading scale: TH. Cycle: A (Second Cycle). Main field: Technology. Language of instruction: The course will be given in English on demand. Optional for: F4, F4bs, Pi4, Pi4bs. Course coordinator: Anders Holst, Anders.Holst@math.lth.se, Numerical Analysis. Recommended prerequisits: FMA021 Applied Mathematics and FMNN10 Numerical Methods for Differential Equations. The course might be cancelled if the number of applicants is less than 15. Assessment: Homework reports and possibly an oral examination. Home page: http://www.maths.lth.se/na/courses/FMN091.

Aim
The aim of the course is to provide a deeper understanding of the construction and application of modern schemes for conservation laws. The focus is on the interaction between mathematical properties of the model and choice of discrete approximation.

Knowledge and understanding
For a passing grade the student must

demonstrate deep knowledge of mathematical and numerical difficulties regarding shock waves, and a deeper understanding of the design and application of modern numerical schemes for nonlinear conservation laws.

Skills and abilities
For a passing grade the student must

- independently be able to select, design and apply advanced numerical methods.

- be able to adapt the methods to varying applications, e.g., shallow water waves, gas dynamics, electromagnetism, ultra sound, etc.

- be able to judge the relevance and accuracy of numerical results.

- report solutions and numerical simulations in written form.

Judgement and approach
For a passing grade the student must

- write a logically well-structured report in suitable terminology on the construction of modern numerical methods and algorithms.

- write an algorithmically well structured report in suitable terminology on the numerical approximation of conservation laws.

Contents
Hyperbolic conservation laws and their properties (weak solution, energy estimates, symmetrization and entropy, shock waves, Riemann problem, Kruzkov solution, stability in L_1, ...). Numerical methods and their stability (upwind-, central-, and relaxation methods, TVD methods and limiters, error estimation using Kruzkov theory). Simulation of shallow water waves and gas dynamics using CLAWPACK.

Literature
1. Randall LeVeque: Finite Volume Methods foir Hyperbolic Problems (ISBN 0 521 00924 3), Cambridge Univ. Press, 2002.
2. Helge Holden and Nils Henrik Risebro: Front Tracking for Hyperbolic Conservation Laws, Springer, New York, 2002.