High-Order Methods for Diffusion Equation with Energy Stable Flux Reconstruction Scheme

Abstract

This study investigates the viscous term formulation for the newly developed energy stable flux reconstruction (ESFR) scheme, as an extension to the linear advection flux formulation in the original ESFR scheme. The concept of energy stability for the flux reconstruction approach is put forward. This paper also discusses the formulation of inviscid first derivative flux based on flux reconstruction and diffusive second derivative flux based on solution reconstruction. Numerical experiments for the linear advection equation, diffusion equation, advection-diffusion equation and viscous Burger’s equations are performed. The stability and accuracy of three recovered schemes, i.e. Nodal Discontinuous Galerkin, Spectral Difference, and Huynh type methods, are studied. Lastly, the choice and implementation of the interface numerical flux are found to affect the stability and accuracy of the various schemes. In particular, central flux is compared with the unbiased upwind and downwind flux. Traditional CFD methods are in general second order accurate schemes. In contrast are high order methods that can achieve arbitrary order of accuracy and produce significant less numerical dissipation than low-order methods. On the other hand, traditional low order schemes are very robust and very simple to implement. They have also been extensively tested and validated in industry. High order methods are in general more complex and less robust. Making the current high order methods simpler, more robust, and more stable is an area of active research. In recent work by Vincent, Castonguay and Jameson, 1 a new class of high-order energy stable flux reconstruction schemes has been developed for linear advection equation. As suggested by its name, this new class of flux reconstruction scheme is energy stable. Various high order methods can be recovered by changing a single parameter in the scheme. This new approach has the potential to offer robustness, efficiency, simplicity and unification tovarious high order schemes, hence a step towards the goal of wide-spread adoption of high order schemes. In this study, we extend the energy stable flux reconstruction scheme to deal with the viscous term by considering the diffusion equation. The rest of the paper is organized as follows. We first discuss the concept of energy stability, and present a brief overview of the energy stability estimate for various high order methods such as nodal Discontinuous Galerkin 2–5 and Spectral Difference. 6–10 In parallel, we outline the the Flux Reconstruction scheme proposed by Huynh 11 and analyze the energy stability estimate for Flux Reconstruction scheme. These set the theme for a short overview of the Energy Stable Flux Reconstruction Scheme in the following section. In greater details, we then present the formulation for the viscous term within the Energy Stable Flux Reconstruction Scheme. This is followed by numerical tests to validate current implementation.