.. _outline:

Outline and Topics
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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