An adaptive wavelet collocation method for the initial value boundary problem of nonlinear PDE’s is studied. The collocation method is based on a cubic spline wavelet decomposition for the Sobolev space H20(I), where I is a bounded interval. Based on a special ‘‘point-wise orthogonality’’ of the wavelet basis functions, a fast discrete wavelet transform (DWT) is constructed. This DWT transform will map discrete samples of a function to its wavelet expansion coefficients in O(N log N) operations. The issue of efficient data structure for the wavelet collocation methods will also be discussed. Numerical results for various PDE’s including wave propagation will be presented.