A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods

Abstract

We introduce a new approach to high-order accuracy for the numerical solution of conservation laws by using flux reconstruction. The approach bridges the two existing approaches of discontinuous Galerkin (DG) and staggered-grid (or spectral difference), and leads to simplified versions of these two methods. It also results in several new schemes with favorable properties. Schemes via the current approach are simple and economical: they solve the conservation laws in differential form and involve only one grid (not two grids that are staggered). In addition, they are conservative. As a result of the new approach, the DG scheme can be formulated using the differential instead of integral form. Another result is a new method, which is similar to the staggered-grid scheme, but more accurate, economical and stable.