Numerical Methods for Hyperbolic Conservation Laws

Tentative plan:

1. Basic theory of hyperbolic conservation laws:
Method of characteristics
Shock formation
Weak solutions
Riemann problem
Euler equation
Shocks and the Hugoniot locus
Rarefaction waves and integral curves

2. Classical numerical methods:
Numerical methods for linear equations
Computing discontinuous solutions
Conservative methods for nonlinear problems
Godunov's method
Approximate Riemann Solvers
Nonlinear stability
High resolution methods
Multidimensional problems (if time permits)

3. Other more advanced topics:
Wave front tracking (in reference [4])
Eno-Weno schemes (in reference [5])
Center di erence scheme (in reference [6])
Scalar conservation laws with discontinuous
ux and applications: selected topics.
Discontinuous Galerkin's method (if time permits)

Possible reading material and text books:

1. R.J. LeVeque, Numerical Methods for Conservation Laws, 2nd ed. Birkauser 1992.
2. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University
Press, 2002.
3. K.W. Morton and D. F. Mayers, Numerical Solution of Partial Di erential Equations.
4. H. Holden, and N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer
Verlag, New York 2002.
5. C.-W. Shu, High order ENO and WENO schemes for computational
uid 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.
6. 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.
7. Class handout of research papers.