Effective high-order energy stable flux reconstruction methods for first-order hyperbolic linear and nonlinear systems

The paper devises high-order energy stable flux reconstruction method towards first-order hyperbolic linear advection-diffusion and nonlinear Navier-Stokes problems (HFR). The underlying idea is to take the advantages of high-order differential framework from flux reconstruction and accurate gradient solution from first-order hyperbolic system. By means of the additional gradient equation, the linear advection-diffusion system is recast as hyperbolic formulation, which is equivalent to original equation when pseudo time derivatives are dropped, and then diffusive fluxes are directly solved by flux reconstruction. The resulting scheme has compact stencils and high accuracy without increasing computation cost. Mathematical analysis is presented to confirm that HFR method is consistent and stable for all orders of accuracy, at least for linear problems. In addition, numerical results for linear and nonlinear problems demonstrate that HFR method can achieve the desired accuracy for both primitive and gradient variable, reduce the absolute error of gradient variable, and allow a larger maximum CFL number compared to the existing methods, such as BR2 and direct flux reconstruction (DFR). It indicates that HFR method is potential in accuracy and efficiency for solving steady viscous problems.