A Simplified Formulation of the Flux Reconstruction Method

The flux reconstruction (FR) methodology has proved to be an attractive approach to obtaining high-order solutions to hyperbolic partial differential equations. However, the utilization of somewhat arbitrarily defined correction polynomials in the application of these schemes, while adding some flexibility, detracts from their ease of implementation and computational efficiency. This paper describes a simplified formuation of the flux reconstruction method that replaces the application of correction polynomials with a single Lagrange interpolation operation. A proof of the algebraic equivalence of this scheme to the FR formulation of the nodal discontinuous Galerkin (DG) method provided that the interior solution points are placed at the zeros of a corresponding Legendre polynomial is presented. Next, a proof of linear stability for this formulation is given. Subsequently, von Neumann analysis is carried out on the new formulation to identify a range of linearly stable schemes achieved by variations of the interior solution point locations. This analysis leads to the discovery of linearly stable schemes with greater formal order of accuracy than the DG method.