Stability of Energy Stable Flux Reconstruction for the Diffusion Problem using the Interior Penalty and Bassi and Rebay II Numerical Fluxes for Linear Triangular Elements

TL;DR

This paper analyzes the stability of energy stable flux reconstruction methods for diffusion problems, deriving conditions for penalty terms with interior penalty and Bassi-Rebay II fluxes, and studying their maximal time steps.

Contribution

It introduces stability conditions for ESFR schemes using IP and BR2 fluxes and shows stability independence from auxiliary correction fields.

Findings

Derived stability conditions for IP and BR2 fluxes.

Proved stability of ESFR schemes is independent of correction fields.

Conducted von Neumann analysis for maximal time step estimation.

Abstract

The flux reconstruction (FR) method has gained popularity within the research community. The approach has been demonstrated to recover high-order methods such as the discontinuous Galerkin (DG) method. Stability analyses have been conducted for a linear advection problem leading to the energy stable flux reconstruction (ESFR) methods also named Vincent-Castonguay-Jameson-Huynh (VCJH) methods. ESFR schemes can be viewed as DG schemes with modally filtered correction fields. Using this class of methods, the linear advection diffusion problem has been shown to be stable using the local discontinuous Galerkin scheme (LDG) to compute the viscous numerical flux. This stability proof has been extended for linear triangular and tetrahedra elements. Although the LDG scheme is commonly used, it requires, on particular meshes, a wide stencil, which raises the computational cost. As a consequence, many prefer the compact interior penalty (IP) or the Bassi and Rebay II (BR2) numerical fluxes. This article, for the first time, derives, for both schemes, a condition on the penalty term to ensure stability. Moreover the article establishes that for both the IP and BR2 numerical fluxes, the stability of the ESFR scheme is independent of the auxiliary correction field. A von Neumann analysis is conducted to study the maximal time step of various ESFR methods.