Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis (AMNA)

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In the lecture the basics of the theory of hyperbolic conservation equations are introduced and then numerical methods for the solution of these equations are discussed.

For the implementation of the practical tasks Matlab or Scilab is recommended. In addition, the program package Clawpack (Conservation Law Package) will be presented with some examples.

The lecture is suitable for students of mathematics as well as for physics.

Topics of the Lecture:

1. Scalar conservation laws
2. Linear hyperbolic systems, examples of nonlinear systems
3. Shock and rarefaction waves, contact discontinuities
4. Numerical methods for linear equations
5. Calculation of unsteady solutions
6. Conservative methods for nonlinear problems
7. The Godunov scheme
8. Approximately solution of the Riemann problem
9. Nonlinear stability
10. Highly accurate methods
11. boundary conditions
12. Kinetic schemes for hyperbolic systems

Literature:

• W. Egartner, Grundlagen der Numerik Physikalischer Erhaltungsgesetze (PDF-Version), Vorlesungsskript, IWR, Universität Heidelberg, 1998.
• R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1990.
• R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, April/Mai 2002.
• C. Schmeiser, Numerische Methoden für hyperbolische Erhaltungssätze, Vorlesungsskript, Institut für Angewandte und Numerische Mathematik, TU Wien.
• J.W. Thomas, Numerical Partial Differential Equation: Conservation Laws and Elliptic Equations, Springer, 1999.

Previous knowledge: Analysis I - III, basic knowledge of ordinary differential equations.

Exercises: Übungsblätter, Materialien

Criteria: Regular participation and participation in the exercise groups, as well as reaching 50% of the possible points on the first seven or the remaining exercise sheets and at least 2/3 of the possible points for the practical tasks.

Additional Literature:

• E. Godlewski und P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996.
• A. Harten, P.D. Lax und B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Review 25 (1983), 35-61.
• R.L. Higdon, Initial-boundary value problems for linear hyperbolic systems, SIAM Review 28 (1986), 177-217.
• A. Jüngel, Modeling and Numerical Approximation of Traffic Flow Problems, Lecture Notes, Universitsät Mainz.
• M. Junk, A. Klar, J. Struckmeier und S. Tiwari, Compact Course Particle Methods for Evolution Equations, AG Technomathematik, Dept. of Mathematics, University of Kaiserslautern 1996.
• D. Kröner, Numerical Schemes for Conservation Laws, John Wiley & Sons, Chichester, 1997.
• R.D. Richtmyer und K.W. Morton, Difference Methods for Initial-Value Problems, Interscience Publishers, 1967.
• C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced numerical approximation of nonlinear hyperbolic equations, Lecture Notes in Mathematics 1697, Springer 1998, 325-432.
• J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, 1983.
• N. Trefethen, Group velocity in finite difference schemes, SIAM Review 24 (1982), 113-136.
• G.B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, Chichester, 1974.