Numerical Solution of Hyperbolic Partial Differential Equations

Lectures begin: Monday 22nd February, 2009.
Exercises begin: Monday 1st March, 2009.

Lecture Notes: Download the lecture notes for the course here (last updated 17/5/2010). Please note that this is not the final version of the notes and there will be changes. Please send an email to harish@math.ethz.ch  or ulrik.fjordholm@sam.math.ethz.ch if you find errors in the notes.

ComPack: Download ComPack 1.02 (Conservation law Matlab Package) here (last updated 17/5/2010).

First exercise class is on Monday 1st March, 2009. You can download exercises and the solutions on this page once they are made available.

Testat Requirement: 70% of exercises reasonably solved.

Exam: Oral examination, 30 min.

Questions? Contact Harish Kumar (harish@math.ethz.ch) 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, WENO, DG and spectral methods in both one- and multi-space dimensions 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, DG and Spectral methods.
• 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, DG and spectral schemes.
• 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 free R2008x Stud).

• 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

Recommended Books:

• 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.

Lecture Notes, Review Articles:

• C. W. Shu, ENO and WENO schemes for hyperbolic conservation laws, NASA, ICASE Report, no 97-65.
• B. Cockburn, Lecture Notes on Discontinuous Galerkin Methods for convection-dominated problems.
  Downloadable from http://www.math.umn.edu/~cockburn/LectureNotes.html
• E. Tadmor, Spectral methods for hyperbolic problems.
  Downloadable from http://www.cscamm.umd.edu/people/faculty/tadmor/pub/subject.htm#Reviews