Lilly, D. K., 1965: On the computational stability of numerical solutions of time-dependent non-linear geophysical fluid dynamics problems. Monthly Weather Review, 93 (1), 11-26.

Abstract: The satisfactory numerical solution of the equations of fluid dynamics applicable to atmospheric and oceanic problems characteristically requires a high degree of computational stability and accurate conservation of certain statistical moments. Methods for satisfying these requirements are described for various systems of equations typical of low Mach number fluid dynamics systems, and are investigated in detail as applied to the two-dimensional, inertial-plane equation for conservation of vorticity in a frictionless non-divergent fluid.
The conservation and stability properties of the spatial differencing methods devised by A. Arakawa are investigated by means of spectral analysis of the stream function into finite Fourier modes. Any of two classes of linear and quadratic conserving schemes are shown to eliminate the non-linear instability discussed by Phillips, although the "aliasing" error remains. Stability related to the time derivative term is investigated through analytic and numerical solutions of a limited-component system of finite spectral equations, equivalent to one of the quadratic conserving difference schemes, and a number of first and second order representations of the time derivative term are tested separately. The commonly used midpoint rule ("leapfrog") method is shown to be unstable in some cases. Of the stable methods, that devised by Miyakoda and the second order Adams-Bashforth method exhibit satisfactory accuracy, while those due to Matsuno, and Lax and Wendroff are much less accurate.
A systematic derivation of the Arakawa difference schemes is contained in an appendix, which shows their unique satisfaction of certain prescribed accuracy and conservation properties.