Numerical Methods for Hyperbolic Partial Differential Equations

Lectures begin: Monday, February 16th
Exercises begin: Monday, February 23rd

ONLINE SUBMISSION OF EXERCISES: http://www.math.ethz.ch/~grsam/submit/?VL=02

Testat Requirement: No testat requirement.

Exam: Oral examination, 30 min.

Lecture notes:
Download the lecture notes for the course here (updated 2015-08-03, see changes).

Compack v1.1B can be downloaded here.

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

Aims of the course

Numerical schemes for hyperbolic equations. 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.
• 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 and Lax-Friedrichs type. Convergence for monotone methods and E-schemes.
• Second-order schemes: Lax-Wendroff, TVD schemes, limiters, strong stability preserving Runge-Kutta methods.
• Linear systems: explicit solutions, energy estimates, first- and high-order finite volume schemes.
• Non-linear Systems: Hugoniot Locus and integral curves, explicit Riemann solutions of shallow-water and Euler equations. Review of available theory.

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. Available online.
  
• E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Ellipses, Paris, 1991.