Outline and Topics

The emphasis of this course is the basic theory of finite difference methods for time-dependent differential equations, including both ODEs and PDEs. The derivation of methods and their analysis in terms of both accuracy and stability are stressed.

• Numerical methods for time-dependent ODEs
  • Taylor series, Runge-Kutta, and Linear Multistep Methods
  • Consistency, order of accuracy, local and global error
  • Zero-stability, A-stability, L-stability, etc.
  • Stiff equations and implicit methods
• Stability theory for PDE methods
  • Method of Lines approach
  • Lax-Richtmyer stability
  • von Neumann stability analysis
  • Relation of ODE and PDE stability theories
• Parabolic PDEs, e.g. diffusion or heat equation
  • Stiffness and the need for implicit solvers
  • Crank-Nicolson method
• Hyperbolic PDEs, e.g. advection and wave equations
  • Lax-Wendroff, upwind methods, etc.
  • Numerical dissipation / dispersion
  • Modified equation analysis
• Mixed equations, e.g. reaction-diffusion, advection-diffusion
  • Fractional step methods
  • Unsplit methods
• Brief introduction to other approaches
  • Finite volume methods
  • Finite element methods
  • Spectral methods