Numerical Methods for Hyperbolic Partial Differential Equations

Lectures begin: Monday, February 17th
Exercises begin: Monday, February 24th

Homework submission:

• Theoretical exercises: submit your homework in the corresponding box in the foyer of rooms HG G 53-54 -- also a write-up/explanation concerning the computer problems should be submitted here
• Computer problems (code and figures):
  • combine your code and figures to a zip-file (Note: you also need to create a zip-file if you only want to submit a single file.) (Everything else like latex-writeups should be submitted as a print-out in the foyer mentioned above.)
  • Go to this page and submit the zip-file.
    
  • You will get a link to a pdf-file, which contains the data of all the files you submitted. Make sure that nothing is missing.
    

No Testat requirement

Homework: Theoretical exercises and programming problems -- biweekly

Exam: Oral examination, 30 min.

Lecture notes can be downloaded here

• Chapter 1 (updated on Mar 5 -- if the document has an older date on its initial page, use the "reloading" button in your browser).
• Chapter 2 (updated on Mar 10)
• Chapter 3 (updated on Mar 31)
• Chapter 4 (updated on Apr 4)
• Chapter 5 (updated on Apr 14)
• Chapter 6 (updated on May 12)
  

Compack v1.1B can be downloaded here (updated on Feb 11).

Questions? Contact Sandra 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 Enginnering.
  
• 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, front tracking, 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 Primer
• 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.
• D. Kroener, Numerical schemes for conservation laws, John Wiley, Stuttgart, 1997.
  
• E. F. Toro, Riemann solvers and numerical methods for fluid dynamics, Springer, Berlin, 2009. Available online.