NTU543 M6660雙曲線型守恆律
NUMERICA L METHODS FOR HYPERBOLIC CONSERVATION LAWS

Dr. Jaw-Yen Yang楊照彥
Institute of Applied Mechanics, National Taiwan University

In this course, we consider the theoretical and computational aspects of hyperbolic conservation laws which originate from the conservation laws of mass, momentum and energy of various fields in science and engineering. For examples, the inviscid Burgers equation, the shallow water equations, the Euler equations that governing the fluid flows. The extension to dissipative transport systems can be done in a straightforward manner. Transport phenomena in modern science and technology are governed by this general class of conservation laws, even the quantum transport in tunneling diode. The theory ranges from classical mechanics, relativity to quantum mechanics. The close relationship between the continuum theory and kinetic theory is also emphasized. Boltzmann-BGK equation based numerical methods will be covered also. Characteristic analysis of the hyperbolic PDE is included. The finite difference (and/or finite volume) method is of main concern. Various standard schemes and their linear stability analysis are introduced. Modern high resolution shock-capturing methods are covered. Computing projects are compulsory and required to help the students to build their own programming capability. Some background in gas dynamics will be useful. This course will be conducted in English.

1. Introduction
2. The Derivation of Conservation Laws
3. Scalar Conservation Laws
4. Nonlinear Systems
5. Linear Hyperbolic Systems
6. Characteristic Analysis
7. The Riemann Problem
8. Numerical Methods for Linear Equations
9. Computing Discontinuous Solutions
10.Conservative Methods for Nonlinear Problems
11. Riemann Solvers
12. Approximate Riemann Solvers
13.Nonlinear Stability
14. High Resolution Methods – TVD, ENO, WENO
15. Modern High-Order Methods – DG & CPR
16. Multi-Dimensional Problems

Text: R. J. LeVeque (1999) Numerical Methods for Conservation Laws, Birkhauser Verlag.

References: Additional class-notes will be provided.

Grading Policy: Homeworks (40%); Computing Projects (30%); Final Exam (30%).