Hybrid flux reconstruction schemes with weighted correction functions

This paper proposes a novel approach to developing hybrid flux reconstruction (HFR) schemes through weighted correction functions. The methodology synthesizes correction functions via linear combinations of existing formulations, leading to enhanced numerical properties. The proposed framework is demonstrated by constructing hybrid schemes based on correction functions from the discontinuous Galerkin (DG), spectral difference (SD), and staggered-grid (SG) schemes, representing three well-established classes of FR schemes. Theoretical investigation through von Neumann analysis reveals the influence of weighting parameters on the dissipation and dispersion characteristics. Numerical experiments with the linear advection equation, linearized Euler equations and nonlinear Euler equations demonstrate that optimally weighted correction functions can achieve superior accuracy and stability compared to their constituent base schemes. Specifically, the optimal weighting coefficient in the HFR1 scheme enables it to achieve even higher accuracy than the DG scheme, whereas the optimal weighting coefficient in the HFR2 scheme yields a CFL limit exceeding that of the SG scheme. From a practical perspective, this work provides a straightforward approach for generating a wide range of FR schemes with predictable numerical characteristics.