Second-Generation Wavelet Collocation Method for the Solution of Partial Differential Equations

An adaptive numerical method for solving partial differential equations is developed. 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 multilevel decomposition, is used for derivative calculations. The treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation 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 complex 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.