On the Adaptive Numerical Solution of Nonlinear Partial Diierential Equations in Wavelet Bases

This work develops fast and adaptive algorithms for numerically solving nonlinear partial di erential equations of the form ut = Lu +N f(u) where L and N are linear di erential operators and f(u) is a nonlinear function. These equations are adaptively solved by projecting the solution u and the operators L and N into a wavelet basis. Vanishing moments of the basis functions permit a sparse representation of the solution and operators. Using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions, are developed for solving evolution equations. For a wavelet representation of the solution u that contains Ns signi cant coe cients, the algorithms update the solution using O(Ns) operations. The approach is applied to a number of examples and numerical results are given. 3