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401-3652-DRL 3 Credits DR D-MATH

Numerical Methods for Hyperbolic Partial Differential Equations (University of Zurich)

Only for ETH D-MATH doctoral students and for doctoral students from the Institute of Mathematics at UZH. They need to register for the I-Math module MAT827DP.
VVZ CR n/a

Last Updated: 2026-02-05 16:37:26

Abstract

This course treats numerical methods for hyperbolic initial-boundary value problems, ranging from wave equations to the equations of gas dynamics. The principal methods discussed in the course are finite volume methods, including TVD, ENO and WENO schemes. 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

* 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.

Resources

Lecture Notes

Lecture slides will be made available to participants. However, additional material might be covered in the course.

Literature

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.

Learning Materials (Links)

General Information

Language
English
Levels
DR
Frequency
Yearly recurring

Examination

Type
ungraded semester performance

Registration & Places

Priority: Registration for the course unit is only possible for the primary target group

Course Components

Type Title Time & Place Hours
lecture Numerical Methods for Hyperbolic Partial Differential Equations (University of Zurich)
**together with University of Zurich**
  • Mon 14:15-16:00 (HG D 3.2)
  • Tue 16:15-18:00 (HG G 26.5)
4 h weekly
exercise Numerical Methods for Hyperbolic Partial Differential Equations (University of Zurich)
**together with University of Zurich**
  • Mon 16:15-17:00 (HG F 3)
1 h weekly

Offered In