Syllabus for Math C42H 3042:
Computational Methods for Gas Dynamics


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