401-3652-00L Numerical Methods for Hyperbolic Partial Differential Equations
|Semester||Spring Semester 2017|
|Lecturers||U. S. Fjordholm|
|Periodicity||yearly recurring course|
|Language of instruction||English|
|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.
|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.
|Prerequisites / Notice||Having attended the course on the numerical treatment of elliptic and parabolic problems is no prerequisite. |
Programming exercises in MATLAB
Former course title: "Numerical Solution of Hyperbolic Partial Differential Equations"