Technical Report 2002-001

Combining Wavelets with Finite Differences: Consistency Analysis

Sônia M. Gomes and Bertil Gustafsson

January 2002

The method is an adaptive finite difference strategy for numerical solution of evolution partial differential equations. The principle is to represent the solution only through those point values indicated by the significant wavelet coefficients. Typically, few points are found in each time step, the grid being coarse in smooth regions, and refined close to irregularities. At each point, the spatial derivatives are discretized by uniform finite differences, using step size proportional to the point local scale. Eventually, required neighboring stencils are not present in the grid. In such case, the correspondig point values are approximated from coarser scales by using reconstruction operators defined by means of interpolating subdivision scheme. Our purpose in this paper is to analyse a generalization of the concept of truncation error, which is the familiar basis of the analysis of difference schemes. For this consistency analysis, we show that the adaptive finite difference scheme can also be formulated in terms of a collocation scheme for an adapted wavelet expansion of the solution. For this purpose, we first prove some results concerning the local behavior of the reconstruction operators, which stand for appropiate cone-like grids.

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