Compact Direct Flux Reconstruction for Conservation Laws

In this study, a high-order discontinuous compact direct flux reconstruction (CDFR) method is developed to solve conservation laws numerically on unstructured meshes. Without explicitly constructing any polynomials, the CDFR method directly calculates the nodal flux derivatives via the compact finite difference approach within elements. To achieve an efficient implementation, the construction procedure of flux derivatives is conducted in a standard element. As a result, a standard flux-derivative-construction matrix can be formulated. The nodal flux derivatives can be directly constructed through the multiplication of this matrix and the flux vectors. It is observed that the CDFR method is identical with the direct flux reconstruction method and the nodal flux reconstruction-discontinuous Galerkin method if Gauss-Legendre points are selected as solution points for degrees p up to 8 tested. A von Neumann analysis is then performed on the CDFR method to demonstrate its linear stability as well as dissipation and dispersion properties for linear wave propagation. Finally, numerical tests are conducted to verify the performance of the CDFR method on solving both steady and unsteady inviscid flows, including those over curved boundaries.