A WAVELET-OPTIMIZED, VERY HIGH-ORDER ADAPTIVE-GRID AND ORDER NUMERICAL-METHOD

Wavelets detect information at different scales and at different locat ions throughout a computational domain. Furthermore, wavelets can dete ct the local polynomial content of computational data. Numerical metho ds are most efficient when the basis functions of the method are simil ar to the data present. By designing a numerical scheme in a completel y adaptive manner around the data present in a computational domain, o ne can obtain optimal computational efficiency. This paper extends the numerical wavelet-optimized finite difference (WOFD) method to arbitr arily high order, so that one obtains, in effect, an adaptive grid and adaptive order numerical method which can achieve errors equivalent t o errors obtained with a ''spectrally accurate'' numerical method.