Midterm

There will be a midterm exam on Wednesday, Feburary 15. The exam will be open book, open notes. The goal will be to insure that you are proficient with some of the theory and algorithms presented in the course.

The focus will be on the following topics:

• Definition of a “hyperbolic” problem (linear and nonlinear)
• Constant coefficient linear systems
  • Diagonalizing a system into decoupled advection equations,
  • Exact solution,
  • Solving the Riemann problem,
  • Phase plane plots for system of 2 equations
  • Interaction of waves from two Riemann problems (e.g. from HW1)
  • Recall that eigenvalues of a triangular matrix are the diagonal elements, and be able to determine eigenvectors, solve linear systems as needed for \(2\times 2\) systems.
• Linearizing a nonlinear problem about a fixed state to obtain a linear system
• Nonlinear scalar problems with convex flux
  • Solving the Riemann problem for shock or rarefaction wave
  • Weak solutions and vanishing viscosity solutions
• Nonlinear systems
  • The Rankine-Hugoniot conditions
  • Determining the Hugoniot locus in a simple case (We will cover this before the midterm.)
• Numerical methods
  • REA Algorithm
  • Basic ideas of Godunov’s method, upwind methods, wave propagation form
  • Idea of introducing slopes and the minmod limiter
  • Definition and importance of TVD methods