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Numerical Solution of Hyperbolic Partial Differential Equations
Numerik der hyperbolischen Differentialgleichungen
Last Updated: 2026-02-05 15:19:48
Abstract
This course treats numerical methods for hyperbolic intial-boundary value problems in one and several space dimensions, ranging from wave equations to the equations of gas dynamics. The principal classes of methods discussed in the course are finite volume methods and discontinuous Galerkin methods. Exercises involve implementation of numerical methods in MATLAB
Objective
The goal of this course is familiarity with the fundamental ideas and mathematical consideration underlying modern numerical methods for conservation laws and wave equations.
Content
1 Scalar linear second-order wave equations 1.1 Wave equations 1.2 Initial and boundary conditions 1.3 Classical and formal solutions 1.4 Domains of dependence and influence 1.5 Weak solutions and abstract wave equations 1.6 Spatial semi-discretization 1.7 Timestepping 1.8 Convergence analysis 1.9 Numerical Dispersion 1.10 Reflections 1.11 Absorbing boundary conditions 2 One-dimensional scalar conservation laws 2.1 Conservation laws 2.2 Characteristics 2.3 Weak solutions 2.4 The Riemann problem 2.5 Entropy conditions 2.6 Properties of entropy solutions 2.7 Supplement: Multidimensional scalar conservation laws 3 Finite volume methods for scalar conservation laws 3.1 Space-time finite differences in 1D 3.2 Finite volume discretization 1D 3.2.1 Consistent numerical flux functions 3.2.2 Godunov's method 3.2.3 Modified equations 3.2.4 Conservation property 3.2.5 Stability 3.2.6 Convergence 3.2.7 Discrete entropy solutions 3.2.8 A priori error estimate 3.2.9 Numerical viscosity 3.3 High resolution methods 3.3.1 Limiters 3.3.2 Central schemes 3.3.3 Method of lines 3.4 Finite volume methods for 2D scalar conservation laws 3.4.1 Operator splitting 3.4.2 Corner transport upwinding 3.4.3 Constant linear advection 3.4.4 Non-constant advection 3.4.5 General conservation laws 3.4.6 2D finite volume methods 4 Galerkin Methods for Scalar Conservation Laws 4.1 Standard Galerkin spatial discretization 4.2 Discontinuous Galerkin methods 4.3 Streamline upwind Petrov Galerkin methods 5 Systems of Conservation Laws in One Space Dimension 5.1 Hyperbolicity 5.2 Linear systems 5.3 The Riemann problem 5.3.1 The linear Riemann problem 5.3.2 Hugoniot loci and shocks 5.3.3 Simple waves and rarefaction 5.4 Entropy conditions 5.5 Multidimensional systems of conservation laws 6 Finite Volume Methods for 1D Systems of Conservation Laws 6.1 Linear systems of conservation laws 6.2 Godunov's method 6.3 Approximate Riemann solvers 6.4 High resolution FVM
Resources
Lecture Notes
Lecture slides wiill be made available to the audience
Literature
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002 D. Kroener: Numerical schemes for conservation laws, Wiley-Teubner, 1997 B. Cockburn: Discontinuous Galerkin Methods for Convection-Dominated Problems, in High Order Methods for Computational Physics, T.J. Barth and H. Deconinck, eds. Springer 1999 E. Tadmor: Entropy stability theory for difference approximations of nonlinear conservation laws and related time dependent problems, Acta Numerica, 2003 M. Feistauer, J. Felcman and I. Straskraba: Mathematical and Computational Methods for Compressible Flow, Clarendon Press 2003
General Information
- Language
- English
- Levels
- BSC , MSC
- Frequency
- Yearly recurring
Examination
- Type
- session examination
- Mode
- oral 30 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture |
Numerik der hyperbolischen Differentialgleichungen
(Räume HG G26.x wegen Umbau geschlossen)
|
|
4 h weekly |
| exercise | Numerik der hyperbolischen Differentialgleichungen |
|
2 h weekly |