Adaptive Multiresolution Collocation Methods for Initial-Boundary Value Problems of Nonlinear PDEs

We have designed a cubic spline wavelet-like decomposition for the Sobolev space $H_0^2 (I)$ where I is a bounded interval. Based on a special point value vanishing property of the wavelet basis functions, a fast discrete wavelet transform (DWT) is constructed. This DWT will map discrete samples of a function to its wavelet expansion coefficients in at most $7N\log N$ operations. Using this transform, we propose a collocation method for the initial boundary value problem of nonlinear partial differential equations (PDEs). Then, we test the efficiency of the DWT and apply the collocation method to solve linear and nonlinear PDEs.