Time Accurate Fast Wavelet-Taylor Galerkin Method for Partial Differential Equations

We introduce the concept of fast wavelet-Taylor Galerkin methods for the numerical solution of partial differential equations. In wavelet-Taylor Galerkin method discretization in time is performed before the wavelet based spatial approximation by introducing accurate generalizations of the standard Euler, and leap-frog time-stepping scheme with the help of Taylor series expansions in the time step. We will present two different time-accurate wavelet schemes to solve the PDEs. First, numerical schemes taking advantage of the wavelet bases capabilities to compress the operators and sparse representation of functions which are smooth, except for in localized regions, up to any given accuracy are presented. Here numerical experiments deal with advection equation with the spiky solution in one dimension, two dimensions, and nonlinear equation with a shock in solution in two dimensions. Second, our schemes deal with more regular class of problems where wavelets are not efficient procedure for data compression but we can use the good approximation properties of wavelet. Here time-accurate schemes lead to consistent mass matrix in an explicit time stepping, which can be solved by approximate factorization techniques. Numerical experiment deals with more regular class of problems like heat equation as well as coupled linear system in two dimensions. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 21: 000–000, 2005