Outline:

1. General properties of partial differential equations. Uniqueness of solutions; method of characteristics; boundary equations appropriate for hyperbolic, parabolic, and elliptic equations.

2. Basic finite difference concepts. Truncation error, consistency, convergence; numerical stability; the Lax-Richtmyer theorem; stability analysis by "direct", "energy", and Fourier method.

3. Stability properties of various time differencing schemes applied to transport and diffusion problems. Euler, Lax-Wendroff, Matsuno, Leapfrog, Adams-Bashforth, DuFort-Frankel, forward-backward, and other schemes; explicit vs. Implicit schemes; effect of spatial differencing on numerical dispersion.

4. Energetically consistent finite difference schemes. Quadratic conservation laws; energy vs. Enstrophy conservation in 2 dimensional flow; the Arakawa Jacqbian; potential vorticity and potential enstrophy conservation in the shallow-water equations.

5. Miscellaneous topics. Dispersion properties of staggered grids; positive-definite and monotonicity-preserving transport schemes; massless coordinate layers; Langrangian advection methods; generalized vertical coordinates.