Graduate Topic Course, Spring 2007

Lectures:
• TR 1:00 -2:15pm, 216 McAllister
Office hours: Monday 9:30-10:30am, Wednesday 9:30-10:30am.
Instructor:
Wen Shen, assistant professor at Department of Mathematics.
Email: shen_w AT math DOT psu DOT edu
Office: 223A McAllister
Home works

Course Description:
This is a topic course for graduate students who are interested in numerical computation. Due to the large applications, this course should be of great interests to students in computer science and engineering etc.

Hyperbolic conservation laws is an important class of nonlinear partial differential equations. The equations arises in modelling whereever some quantities are preserved, such as mass, momentum and energy. These equations are of importance to a broad spectrum of disciplines, such as: gas dynamics, fluid mechanics, elasticity and visco-elasticity, chromatography, trafic flow, geophysics, meteorology, electromagnetism, astrophysics, etc.

Due to the nonlinearity, solutions to these equations will become discontunuous in finite time even with smooth initial data. They are called shock waves. The presence of shock waves accounts for most of the difficulties in the theoretical and numerical studies of these problems. In particular, the most natural numerical schemes based on the approximation of high order derivatives cannot be implemented here, due to lack of smoothness. Therefore, a standard course on numerical methods for partial differential equations would not be able cover this topic in decent depth.

Numerical methods for hyperbolic conservation laws has been an extremely active research field in the past few decades. Several classes of methods were developed. The most popular methods are of finite difference type, adapted especially to capture discontinuities. Many other methods are also available, including wave front tracking, finite element, finite volume, and spectrum methods etc.

I plan to use the first 1/4 of time to go through some basic theoretical results on conservation laws, and the last 3/4 on numerical methods. The students do not need to have extensive background knowledge on partial differential equation. I will cover some theories that are needed. You should know you calculus with one- and multi-variables. Some programming background and basic knowledge on numerical analysis will be helpful.

I plan to cover the following topics:

From the book of LeVeque (reference [1]):
• 1 -- Basic theories on hyperbolic conservation laws (3 --> -- weeks)
• * method of characteristics
• * shock formation
• * Weak solutions
• * Riemann problem
• * Euler equation
• * Shocks and the Hugoniot locus
• * Rarefaction waves and integral curves
• 2 -- Classical numerucal methods: (6-8 weeks)
• * Numerical methods for linear equations
• * Computing Discontinuous solutions
• * Conservative methods for nonlinear problems
• * Godnov's method
• * Approximate Riemann Solvers
• * Nonlinear stability
• * High resolution methods
• * multidimensional problems
• 3 -- Other more recent topics (the remaining time)
• * Eno-Weno schemes (in reference [3])
• * center difference scheme of Tadmor (in reference [4])
• * Wave front tracking (in reference [2])
• * discontinuous Galerkin's method (time dependent)
Possible reading material and text books:
• [1] R.J. LeVeque, Numerical Methods for Conservation Laws, 2nd ed. Birkauser 1992.
• [2] H. Holden, and N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer Verlag, New York 2002.
• [3] C.-W. Shu, High order ENO and WENO schemes for computational fluid dynamics, in "High-Order Methods for Computational Physics", T.J. Barth and H. Deconinck, editors, Lecture Notes in Computational Science and Engineering, volume 9, Springer, (1999), 439-582.
• [4] E. Tadmor, Approximate solutions of nonlinear conservation laws, in "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations", Lecture notes in Mathematics 1697, 1997 C.I.M.E. course in Cetraro, Italy, June 1997 (A.~Quarteroni ed.) Springer Verlag 1998, 1-149.
• [5] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
• + class handout of research papers.
Book [1] will be heavily used. I recommend you to buy it.
Other materials will be handed out in class.
Last updated Oct 20, 2006 by Wen Shen.