ON THE ADAPTIVE NUMERICAL-SOLUTION OF NONLINEAR PARTIAL-DIFFERENTIAL EQUATIONS IN WAVELET BASES

This work develops fast and adaptive algorithms for numerically solvin g nonlinear partial differential equations of the form u(t) = Lu + Nf( u), where L and N are linear differential operators and f(u) is a nonl inear function. These equations are adaptively solved by projecting th e 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 ad aptive algorithms that apply operators to functions and evaluate nonli near functions, are developed for solving evolution equations. For a w avelet representation of the solution u that contains N-s significant coefficients, the algorithms update the solution using O(N-s) operatio ns. The approach is applied to a number of examples and numerical resu lts are given. (C) 1997 Academic Press.