Smoothing, filtering, and boundary effects

Numerical integrations of finite‐difference analogs of systems of nonlinear partial differential equations, such as those arising in atmospheric dynamics, are subject to computational instability from a variety of causes. One type of instability is produced by a spurious, nonlinear growth of high‐frequency components that may be introduced by roundoff, truncation, and observational error. This type of instability, first discussed by N. A. Phillips, can be suppressed by a suitable choice of finite‐difference method or by the use of a filter that selectively damps the high‐frequency components. Though much effort is being devoted to the development of stable finite‐difference procedures, and considerable success has been achieved, all such methods involve high‐frequency smoothing either implicitly or explicitly. It is therefore important that the effects of such filtering be fully understood. Filtering and smoothing operators are developed for use in conjunction with the numerical integration of nonlinear systems and for other purposes. The general procedure is demonstrated for simple one‐dimensional operators and the properties of such operators are thoroughly explored. The development is then expanded to allow for compound operators designed to suit some particular requirement and further extended to more than one dimension. Both real and complex operators are discussed. Reverse smoothers or wave amplifiers are introduced, and some of the problems associated with their use are discussed. A general procedure is outlined for the construction of an ‘ideal’ low‐pass filter; that is, a filter that removes the shortest resolvable wave component (the 2‐grid‐interval wave) but restores all other wave components as close as is desired to their original amplitudes without amplifying or changing the phase of any wave component. Finally, the effects, sometimes disastrous, of finite domains on the properties of the smoothing operators are explored for a variety of common boundary assumptions.