Figure 1: The initial condition for variable h at four different wavelet thresholds.
These days there is a tremendous focus on building adaptive numerical methods in the computational sciences. The simple reason is that interesting physical problems rarely occur with uniform scale features throughout the computational domain. We can, of course, apply uniform grid techniques to such problems with a tremendous waste of computational effort. In fact, it is likely that a well-written adaptive scheme run on a five year old, or older, supercomputer can outperform a nonadaptive scheme on the current generation of supercomputers. It is this kind of savings that must be obtained in order to conduct leading edge oceanography.
Now a few comments on wavelet-based numerical methods. Generally, one can classify “wavelet methods” as either a collocation type or a Galerkin type. One can think of the computational parameters in a wavelet collocation method as the point values of the computational variables in the physical space. Likewise, one can think of the computational parameters in a wavelet Galerkin method as the point values in the transform space, in this case the wavelet transform space. In either case, one is working with N real numbers. The best way to evolve these N real numbers in a wavelet or multiresolution framework is a matter of debate. This paper will introduce a numerical method named the Wavelet-Optimized Finite Difference (WOFD) method.
Explanations of WOFD have appeared in many places, see [9], [12] and [8]. The first occurrence of the argument that wavelets should be used only to analyze computational data for error detection and grid generation occurred in [9] and subsequently in [12].
