A Dynamically Adaptive Multilevel Wavelet Collocation Method for Solving Partial Differential Equations in a Finite Domain

Liandrat and Tchiamichian [2], Bacry et al. [3], Maday and Ravel [4], and Bertoluzza et al. [5] have shown that A dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations. The the multiresolution structure of wavelet bases is a simple multilevel structure of the algorithm provides a simple way to adapt and effective framework for spatially adaptive algorithms. computational refinements to local demands of the solution. High In their Galerkin algorithms, they retain wavelets, whose resolution computations are performed only in regions where sharp coefficients are larger than a given threshold. In order to transitions occur. The scheme handles general boundary condibe able to track singularities they also retain wavelets that tions. The method is applied to the solution of the one-dimensional Burgers equation with small viscosity, a moving shock problem, are adjacent to such regions. This adaptive procedure, and a nonlinear thermoacoustic wave problem. The results indicate based on the analysis of wavelet coefficients, allows them that the method is very accurate and efficient. Q 1996 Academic to follow the local structures of the solution. Press, Inc. In wavelet Galerkin algorithms nonlinearities can be handled using either the connection coefficients (see [3]) introduced by Beylkin [6, 7] or quadrature formulae (see