Performance and accuracy of hybridized flux reconstruction schemes

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We present a unified framework for the hybridization of flux reconstruction (FR) schemes. Specifically, we discuss the hybridized flux reconstruction (HFR) and interior-embedded flux reconstruction (EFR) methods and analyze their performance and accuracy for a range of correction functions on quadrilateral elements. HFR uses discontinuous traces on the faces of the elements, and EFR considers globally continuous trace functions. We perform two-dimensional von Neumann analysis to characterize the numerical error of EFR methods in relation to HFR and FR. HFR and FR are equivalent for linear advection, but EFR is only equivalent to FR one-dimensional advection directions. It is observed that EFR methods generally introduce additional numerical error for lower-order simulations, but behave closer to FR methods for higher orders. In addition, the behavior for different correction functions follows the same trend as the conventional FR formulation. Verification with linear and nonlinear numerical examples showed the expected order of accuracy of p+1 was obtained for all formulations, where p is the polynomial degree of the solution. With appropriate choices of correction functions, EFR methods can be more accurate than other HFR and FR methods. Furthermore, it is observed that the choice of correction function has an impact on the performance of the implicit solver by a factor of up to 2 for FR methods for an airfoil-vortex interaction problem, but had a smaller influence on the hybridized forms. For this problem, we observed speedups factors of up to 4.3 and 6.7 for HFR and EFR, respectively, when compared with FR. Hence, hybridized methods are a viable approach to reducing the computational cost of implicit FR solvers.