Popular high-order schemes with compact stencils for Computational Fluid Dynamics (CFD) include Discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV) methods. The recently proposed Flux Reconstruction (FR) approach or Correction Procedure using Reconstruction (CPR) is based on a differential formulation and provides a unifying framework for these high-order schemes. Here we present a brief review of recent developments for the FR/CPR schemes as well as some pacing items.
IntroductionIn the field of Computational Fluid Dynamics (CFD), low -order methods are generally robust and reliable; as a result, they are routinely employed in practical calculations. For the same computing cost, high-order methods can provide considerably more accurate solutions, but they are more complicated and less robust. The need to improve and develop new high-order methods with favorable properties has attracted the interest of many researchers as evidenced by the recently held First (2012) and Second (2013) International Workshops on High-Order CFD Methods.The Discontinuous Galerkin (DG) method is currently among the most widely used high-order numerical methods for solving the compressible Navier-Stokes equations on unstructured meshes. It was introduced for the neutron transport equation by Reed and Hill (1973), analyzed by LaSaint and Raviart (1974) and developed and made popular for fluid dynamics equations by Cockburn,
The flux reconstruction (FR) method offers a simple, efficient, and easy to implement method, and it has been shown to equate to a differential approach to discontinuous Galerkin (DG) methods. The FR method is also accurate to an arbitrary order and the isentropic Euler vortex problem is used here to empirically verify this claim. This problem is widely used in computational fluid dynamics (CFD) to verify the accuracy of a given numerical method due to its simplicity and known exact solution at any given time. While verifying our FR solver, multiple obstacles emerged that prevented us from achieving the expected order of accuracy over short and long amounts of simulation time. It was found that these complications stemmed from a few overlooked details in the original problem definition combined with the FR and DG methods achieving high-accuracy with minimal dissipation. This paper is intended to consolidate the many versions of the vortex problem found in the literature and to highlight some of the consequences if these overlooked details remain neglected.
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