Figure 2: The initial wavelet generated grids at four different wavelet thresholds. From top to bottom the four different wavelet thresholds are .001, .0001, .00001, and 0.
Wavelet analysis provides information on the energy present at various scales and locations throughout a computational domain. This information is precisely the information that is needed to define the appropriate grid point densities and the appropriate numerical order to resolve the physics at hand in the computationally most efficient manner. Here we apply the numerical method known as the Wavelet Optimized Finite Difference Method (WOFD) to a model problem in oceanography. WOFD is a completely dynamically adaptive numerical method which has the ability to focus on small scale physics throughout the computational domain as the scale of the physics gets smaller and is transported across the domain. In this manner, WOFD applies the computational effort where it is needed without overcomputing in the regions of the domain where the physics is somewhat smooth and perhaps more linear. The version of WOFD which is used here is such that both the spatial and temporal orders will be fixed at four. In the areas of oceanography and climate modeling, order four can be considered high order and we will therefore call the method the Adaptive High Order (AHO) version of WOFD, or simply WOFD-AHO.
In our four figures we show the state of one flow variable in Figures (1) and (3) and the corresponding wavelet generated grids in Figures (2) and (4). Generally one can obtain a very large computational savings by using such wavelet generated grids.
3 A New Assimilation Technique Using Wavelet Analysis: Jameson and Waseda
Wavelet analysis provides a mechanism to detect errors throughout a computational domain. This is possible since scaling functions are constructed in order to approximate low order polynomials exactly up to a given order and wavelets can detect deviation from these low-order polynomials. This is exactly the kind of information which is needed in order to estimate computational error. We will illustrate the use of wavelet analaysis in a new technique to assimulate data into computational model. The new method is fast and efficient and we believe that it surpasses other methods of comparable computational cost.
