High-Order Flux Reconstruction Schemes

There is an increasing desire among industrial practitioners of computational fluid dynamics to undertake high-fidelity scale-resolving simulations of unsteady flows within the vicinity of complex geometries. Such simulations require numerical methods that can operate on unstructured meshes with low numerical dissipation. The flux reconstruction (FR) approach describes one such family of numerical methods, which includes a particular type of collocation-based nodal discontinuous Galerkin method, and spectral difference methods, as special cases. In this chapter we describe the current state-of-the-art surrounding research into FR methods. To begin, FR is described in one dimension for both advection and advection–diffusion problems. This is followed by a description of its extension to multidimensional tensor product and simplex elements. Stability and accuracy issues are then discussed, including an overview of energystability proofs, von Neumann analysis results, and stability characteristics when the flux function of the governing system is nonlinear. Finally, implementation aspects Handbook of Numerical Analysis, Vol. 17. http://dx.doi.org/10.1016/bs.hna.2016.09.010 © 2016 Elsevier B.V. All rights reserved. 227 are outlined in the context of modern hardware platforms, and three example applications of FR are presented, demonstrating the potential utility of FR schemes for scale resolving simulation of unsteady flow problems.