```
```
Syllabus for Math C42H 3042:
Computational Methods for Gas Dynamics
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0. Introduction
• Eulers equations and conservation laws
• Overview of CFD

1. Scalar, Linear Problems
1. Exact solutions
• Linear advection equation, Cauchy problem
• Characteristics, domain of dependence
• Riemann problem
• Vanishing viscosity solutions

2. 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

3. 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
1. 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

2. Numerical methods
• Conservative schemes and weak solutions
• Lax-Wendroff theorem:
For conservative methods, Ul 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
1. Exact behaviour
• Characteristics, Riemann problem
• Primitive variables

2. Numerical methods
• First order Godunov and the entropy condition
• Roe's approximate Riemann solver
• Higher order Godunov
• Boundary conditions
• Multidimensions