Submissions1

1. Adaptive wavelet representation and differenciation on block-structured grids

Margarete. O. Domingues
Instituto Nacional de Pesquisas Espaciais - LMO/CPTEC
margaret@cptec.inpe.br

This paper considers an adaptive finite difference scheme for the numerical solution of evolution partial differencial 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 dinamically 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.

Magda. K. Kaibara
Universidade Estadual Paulista - Depto. de Matemática
kaibara@fc.unesp.br

Sônia M. Gomes
Universidade Estadual de Campinas - IMECC
soniag@ime.unicamp.br

The scheme is based on Ami Harten's ideas, the main tools coming from wavelet theory, in the framework of multiresolution analysis for cell averages. But instead of evolving cell averages on the finest uniform level, we propose to evolve the cell averages on the grid determined by the significant wavelet coefficients. Typically, there are few cells in each time step, big cells on smooth regions, and smaller ones close to irregularities of the solution. For the numerical flux, we use a simple uniform central finite difference scheme, adapted to the size of each cell. If any of the required neighboring cell averages is not present, it is interpolated from coarser scales. But we switch to ENO scheme in the finest part of the grids. To show the feasibility and efficiency of the method, it is applied to a system arising in polymer-flooding of an oil reservoir. In terms of CPU time and memory requirements, it outperforms Harten's multiresoltution algorithm.

3. Implicit Discontinuous Galerkin Method With Wavelet-Based Space Adaptivity

Sônia M. Gomes
Universidade Estadual de Campinas - IMECC
soniag@ime.unicamp.br

In this paper, adaptivity in space is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the Discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. This kind of information is also used in adaptive choices of the artificial diffusion coefficient and the kind of numerical flux adopted on cells' interfaces. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method.

4. Stability Analysis of a Multilevel Wavelet Scheme for Evolution Equations with Time Adaptivity

José. E. Castilho
Universidade Federal de Uberlândia -Faculdade de Matemática
jecastilho@ufu.br

Sônia M. Gomes
Universidade Estadual de Campinas -IMECC
soniag@ime.unicamp.br

The study in this paper is devoted to the stability and consistency analyses of an adaptive multilevel time discretization proposed by Bacry, Mallat and Papanicolau. The main idea is to evolve the components in a multirresolution representation of the numerical solution by means of an explicit algorithm, adapting the time step according to each scale level. For a model problem, and in the context of biorthogonal wavelets, it is proved that the stability condition and consistency order are the same as in the original non-adapted scheme.