Numerical Solution of Hyperbolic Partial Differential Equations

Lectures begin: Monday 27st February, 2012.
Exercises begin: Monday 5th March, 2012.

Testat Requirement: 70% of exercises reasonably solved.

Exam: Oral examination, 30 min.

Lecture notes: Download the lecture notes for the course here (last updated 19/3/2012). Notes on high-order accurate schemes here (last updated 2/5/2012).

Compack can be downloaded here.

Questions? Contact Ulrik in case of questions concerning the exercises or the lecture.

Aims of the course

Numerical schemes for Hyperbolic equations, particularly systems of equations like the Euler equations of gas dynamics will be presented. Finite volume schemes, TVD, ENO and WENO will also be described. Practical exercises will involve Matlab implementation of the numerical methods.

Content of the course

• Introduction to Hyperbolic problems: Conservation, flux modeling, examples and significance in Physics and Engineering.
• Linear Advection equations in one dimension: Characteristics, Energy estimates, upwind schemes, variable co-efficients.
• Scalar conservation laws: Shocks, Rarefactions, solutions of the Riemann problem, weak and entropy solutions, Some existence and uniqueness results, Finite volume schemes of the Godunov, Engquist-Osher, Lax-Friedrichs type, Convergence for Monotone and E-schemes.
• Second-order schemes: Lax-Wendroff, TVD schemes, Limiters, Strong-Stability preserving Runge-Kutta schemes.
• High-order schemes: ENO, WENO.
• Linear systems: Explicit solutions, Energy estimates, First- and high-order finite volume schemes.
• Non-linear Systems: Examples, General concepts: Hugoniot Locus and Integral curves, Explicit Riemann solutions of Shallow-water and Euler equations. Review of available theory.
• First-Order finite volume schemes for systems: Godunov flux, HLL and Roe solvers, examples of Shallow-water and Euler equations, Entropy stability.
• High-order schemes for systems: TVD, ENO, WENO.
• Hyperbolic equations in multi-dimensions: Theory, Finite-volume schemes on Cartesian meshes, Dimensional splitting.
• Schemes on Unstructured meshes: Review of mesh generation, FV and DG schemes on unstructured meshes. Examples of Shallow-water and Euler equations with complex geometry.
• Possible Advanced Topics: Magneto-HydroDynamics (MHD), Adaptive Mesh Refinement, Incompressible flows.

Matlab links

ETH students can download Matlab with a free network license from Stud-IDES.

• Matlab Online Documentation von MathWorks
• Matlab Primer
• Matlab Summary and Tutorial
• Matlab Online Documentation from the MathWorks
• Matlab Guide, D.J. Higham and N.J. Higham, SIAM 2000, ISBN 0-89871-516-4
• Matlab Einführungskurs, P. Arbenz

Literature

Related books, Lecture Notes, Review Articles:

• H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer 2011. Available online.
  
• R . J. LeVeque, Finite Volume methods for hyperbolic problems, Cambridge university Press, 2002.
• E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Ellipses, Paris, 1991.
• D. Kroner, Numerical schemes for Conservation laws, John Wiley, Stuttgart, 1997.
• E. F. Toro, Riemann Solvers and Numerical methods for fluid dynamics, Springer, Berlin, 1997.