Day/time: Wednesday, 1:30-4:30 pm
Start date: January 9- April 10, 2002
Location: The Fields Institute

This course will cover basic techniques for solving partial differential equations on the computer, with emphasis on finite difference methods. Special attention will be paid to how the features of a good discretization reflect the mathematical properties of the PDE being solved.

Topics

• Parabolic equations
  • Explict and implicit discretizations
  • Consistency, stability, and convergence
  • Von Neumann stability (Fourier analysis)
  • Variational inequalities and free boundaries
  • Multi-dimensional problems
• Elliptic equations
  • The maximum principle
  • Solution of sparse linear systems
• Hyperbolic equations and conservation laws
  • CFL stability
  • Flux conservation and shock capturing
• Variational formulations and finite element methods
• Possible special topics (if time permits)
  • The Euler and Navier-Stokes equations of fluid dynamics
  • Adaptive mesh refinement techniques
  • Spectral and pseudo-spectral methods
  • Special techniques: random walkers, lattice gas