Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement

摘要

We present a WENO finite volume method for the approximation of hyperbolic conservation laws on adaptively refined Cartesian grids.On each single patch of the AMR grid, we use a modified dimension-by-dimension WENO method, which was recently developed by Buchmüller and Helzel (2014) [1]. This method retains the full spatial order of accuracy of the underlying one-dimensional WENO reconstruction for nonlinear multidimensional problems, and requires only one flux computation per interface. It is embedded into block-structured AMR through conservative interpolation functions and a numerical flux fix that transfers data between different levels of grid refinement.Numerical tests illustrate the accuracy of the new adaptive WENO finite volume method. Compared to the classical dimension-by-dimension approach, the new method is much more accurate while it is only slightly more expensive. Furthermore, we also show results of an accuracy study for an adaptive WENO method which uses multidimensional reconstruction of the conserved quantities and a high-order quadrature formula to compute the fluxes. While the accuracy of such a method is comparable with our new approach, it is about three times more expensive than the latter.