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


Prof. Dr. Matthias Ehrhardt

Lecture Winter Term 2018/19:

Numerical Methods for Hyperbolic Problems
(Advanced Topic)

(Start of Lecture 15.10.18)
 Lecture   Mo,  16:00 - 17:30   Room G.14.34 
   Tu,  14:15 - 15:45   Raum G.15.20 

Outline of the Lecture

Hyperbolic conservation laws are (mostly nonlinear) first order partial differential equations that describe the temporal evolution of transport processes. These are mostly balance equations for densities of physical conserved quantities (mass, momentum, energy). The most important application example is the Euler equations of gas dynamics, e.g. for the flow around wings or vehicles. Other applications include the modeling of shallow water waves, traffic flows, climate models, multiphase flow, and magnetohydrodynamics.

The challenge in the theoretical and numerical treatment of hyperbolic conservation equations lies in the fact that solutions of conservation equations usually form discontinuities after a certain time (even with smooth initial data!), And therefore suitable weak solutions and numerical methods have to be considered.

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
detailed outline of the lecture.


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:

University of Wuppertal
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Applied Mathematics & Numerical Analysis Group

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