Adaptive wavelet representation and differentiation of block-structured grids

This paper considers an adaptive finite difference scheme for the numerical solution of evolution partial differential equations. The computational domain is formed by non-overlapping blocks. Each block is a uniform grid, but step size may change from one block to another. The blocks are not predetermined, but they are dynamically constructed according to the refinement needs of the numerical solution. The decision over whether a block should be refined or unrefined is taken by looking at the magnitude of wavelet coefficients of the numerical solution on such block. The main objective of this paper is to establish a general framework for the construction and operation on such adaptive block-grids in 2D. The algorithms and data structure are formulated by using abstract concepts borrowed from quaternary trees. This procedure helps the understanding of the method and its computational implementation. The ability of the method is demonstrated by solving some typical test problems.