An adaptive multiresolution method on dyadic grids: Application to transport equations

Abstract

We propose a modified adaptive multiresolution scheme for solving d-dimensional hyperbolic conservation laws which is based on cell-average discretization in dyadic grids. Adaptivity is obtained by interrupting the refinement at the locations where appropriate scale (wavelet) coefficients are sufficiently small. One important aspect of such a multiresolution representation is that we can use the same binary tree data structure for domains of any dimension. The tree structure allows us to succinctly represent the data and efficiently navigate through it. Dyadic grids also provide a more gradual refinement as compared with the traditional quad-trees (2D) or oct-trees (3D) that are commonly used for multiresolution analysis. We show some examples of adaptive binary tree representations, with significant savings in data storage when compared to quad-tree based schemes. As a test problem, we also consider this modified adaptive multiresolution method, using a dynamic binary tree data structure, applied to a transport equation in 2D domain, based on a second-order finite volume discretization.