Numerical Methods for Hyperbolic Conservation Laws

Lecture Notes

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.