Syllabus for Math C42H 3042:

Computational Methods for Gas Dynamics

0. Introduction

- Eulers equations and conservation laws
- Overview of CFD

1. Scalar, Linear Problems

- Exact solutions

- Linear advection equation, Cauchy problem
- Characteristics, domain of dependence
- Riemann problem
- Vanishing viscosity solutions

- Numerical solutions for smooth data

- Finite difference and finite volume
- Basic schemes and stencils
- LTE and consistency, order of accuracy
- Global error, norms and convergence
- Stability
- Lax Equivalence Theorem:

For consistent linear schemes, stability implies convergence- Analysis of example schemes
- von Neumann stability analysis
- CFL condition
- Upwinding systems
- geometric flux interpretation and conservative schemes

- Discontinuous data

- Example "jump" advection, diffusion & oscillation behaviour
- Modfied equations and phase error
- Godunov's theorem:

For consistent linear schemes, max-norm bounded implies 1st order- TVD schemes
- Hybrid nonlinear scheme for linear problem
- Slope limiters

2. Nonlinear, scalar equations

- Exact behaviour

- Flux functions (convex)
- chacteristics
- shock formation
- Burger's equation
- weak solutions (conservation) and shock speed
- Riemann problems, rarefaction, sonic points
- Entropy condition
- Other examples

- Numerical methods

- Conservative schemes and weak solutions
- Lax-Wendroff theorem:

For conservative methods, U_{l}converges implies the limit is a weak solution.- Entropy conditions
- Godunov's method
- Non-linear stability analysis:

For conservative methods, TV-stable implies convergence.

3. Nonlinear, hyperbolic systems and Euler equations

- Exact behaviour

- Characteristics, Riemann problem
- Primitive variables

- Numerical methods

- First order Godunov and the entropy condition
- Roe's approximate Riemann solver
- Higher order Godunov
- Boundary conditions
- Multidimensions