Method for the Solution of Partial Differential Equations

An adaptive numerical method for solving partial differential equations is devel-oped. The method is based on the whole new class of second-generation wavelets. Wavelet decomposition is used for grid adaptation and interpolation, while a new O(N) hierarchical finite difference scheme, which takes advantage of wavelet mul-tilevel decomposition, is used for derivative calculations. The treatment of nonlinear terms and general boundary conditions is a straightforward task due to the col-location nature of the algorithm. In this paper we demonstrate the algorithm for one particular choice of second-generation wavelets, namely lifted interpolating wavelets on an interval with uniform (regular) sampling. The main advantage of using second-generation wavelets is that wavelets can be custom designed for com-plex domains and irregular sampling. Thus, the strength of the new method is that it can be easily extended to the whole class of second-generation wavelets, leaving the freedom and flexibility to choose the wavelet basis depending on the application. c © 2000 Academic Press Key Words: wavelets; lifting scheme; second-generation wavelets; partial differ-ential equations; adaptive grid; numerical method. 1