Wavelets detect information at different scales and at different locations throughout a computational domain. Furthermore, wavelets can detect the local polynomial content of computational data. Numerical methods are most efficient when the basis functions of the method are similar to the data present. By designing a numerical scheme in a completely adaptive manner around the data present in a computational domain, one can obtain optimal computational efficiency. This paper extends the numerical wavelet-optimized finite difference (WOFD) method to arbitrarily high order, so that one obtains, in effect, an adaptive grid and adaptive order numerical method which can achieve errors equivalent to errors obtained with a "spectrally accurate" numerical method. [References: 43]
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