High-order adaptive finite-volume schemes in the context of multiresolution analysis for dyadic grids

Abstract

This paper considers the design of adaptive finite-volume discretizations for conservation laws. The methodology comes from the context of multiresolution representation of functions, which is based on cell averages on a hierarchy of nested grids. The refinement process is performed by the partition of each cell at a certain level into two equal child cells at the next refined level by a hyperplane perpendicular to one of the coordinate axes, which varies cyclically from level to level. The resulting dyadic grids allow the organization of the multiscale information by the same binary-tree data structure for domains in any dimension. Cell averages of neighbouring stencil cells, chosen on the subdivision direction axis, are used to approximate the cell average of the child cells in terms of a classical A. Harten prediction formula for 1D discretizations. The difference between successive refinement levels is encoded as the prediction errors (wavelet coefficients) in one of the child cells. Adaptivity is obtained by interrupting the refinement at the cells where the wavelet coefficients are sufficiently small. The efficiency of the adaptive method is analysed in applications to typical test problems in one and two space dimensions for second- and third-order schemes for the space discretization (WENO) and time integration (explicit Runge–Kutta). The results show that the adaptive solutions fit the reference finite-volume solution on the finest regular grid, and memory and CPU requirements can be considerably reduced, thanks to the efficient self-adaptive grid refinement.