A Fast Adaptive Wavelet Collocation Algorithm for Multidimensional PDEs

Abstract

A fast multilevel wavelet collocation method for the solution of partial differential equations in multiple dimensions is developed. The computational cost of the algorithm is independent of the dimensionality of the problem and is O(N), where N is the total number of collocation points. The method can handle general boundary conditions. The multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution. High resolution computations are performed only in regions where singularities or sharp transitions occur. Numerical results demonstrate the ability of the method to resolve localized structures such as shocks, which change their location and steepness in space and time. The present results indicate that the method has clear advantages in comparison with well established numerical algorithms. © 1997 Academic Press.