An Adaptive Pseudo-Wavelet Approach for Solving Nonlinear Partial Differential Equations

Abstract We numerically solve nonlinear partial differential equations of the form u t = ℒ u + N f u , where ℒ and N are linear differential operators and f ( u ) is a nonlinear function. Equations of this form arise in the mathematical description of a number of phenomena including, for example, signal processing schemes based on solving partial differential equations or integral equations, fluid dynamical problems, and general combustion problems. A generic feature of the solutions of these problems is that they can possess smooth, nonoscillatory and/or shock-like behavior. In our approach we project the solution u ( x, t ) and the operators ℒ and N into a wavelet basis. The vanishing moments of the basis functions permit a sparse representation of both the solution and operators, which has led us to develop fast, adaptive algorithms for applying operators to functions, e.g. , ℒ u, and computing functions, e.g. , f ( u )  = u 2 , in the wavelet basis. These algorithms use the fact that wavelet expansions may be viewed as a localized Fourier analysis with multiresolution structure that is automatically adaptive to both smooth and shock-like behavior of the solution. In smooth regions few wavelet coefficients are needed, and in singular regions large variations in the solution require more wavelet coefficients. Our new approach allows us to combine many of the desirable features of finite-difference, (pseudo) spectral and front-tracking or adaptive grid methods into a collection of efficient, generic algorithms. It is for this reason that we term our algorithms as adaptive pseudo-wavelet algorithms. We have applied our approach to a number of example problems and present numerical results.