An Analysis of Stability of the Flux Reconstruction Formulation on Quadrilateral Elements for the Linear Advection–Diffusion Equation

The Flux Reconstruction (FR) approach to high-order methods is a flexible and robust framework that has proven to be a promising alternative to the traditional Discontinuous Galerkin (DG) schemes on parallel architectures like Graphical Processing Units (GPUs) since it pairs exceptionally well with explicit time-stepping methods. The FR formulation was originally proposed by Huynh (AIAA Pap 2007-4079:1–42, 2007). Vincent et al. (J Sci Comput 47(1):50–72, 2011) later developed a single parameter family of correction functions which provide energy stable schemes under this formulation in 1D. These schemes, known as Vincent–Castonguay–Jameson–Huynh (VCJH) schemes, offer control over properties like stability, dispersion and dissipation through the variation of the VCJH parameter. Classical schemes like nodal-DG and Spectral Difference (SD) can also be recovered under this formulation. Following the development of the FR approach in 1D, Castonguay et al. (J Sci Comput 51(1):224–256, 2012) and Williams et al. (J Comput Phys 250:53–76, 2013) and Williams and Jameson (J Sci Comput 59(3):721–759, 2014) developed correction functions that give rise to energy stable FR formulations for triangles and tetrahedra. For the case of tensor product elements like quadrilaterals and hexahedra however, a simple extension of the 1D FR approach utilizing the 1D VCJH correction functions was possible and has been adopted by several authors (Castonguay in High-order energy stable flux reconstruction schemes for fluid flow simulations on unstructured grids, 2012; Witherden et al. in Comput Fluids 120:173–186, 2015; Comput Phys Commun 185(11):3028–3040, 2014). But whether such an extension of the 1D approach to tensor product elements is stable remained an open question. A direct extension of the 1D stability analysis fails due to certain key difficulties which necessitate the formulation of a norm different from the one utilized for stability analysis in 1D and on simplex elements. We have recently overcome these issues and shown that the VCJH schemes are stable for the linear advection equation on Cartesian meshes for any non-negative value of the VCJH parameter (Sheshadri and Jameson in J Sci Comput 67(2):769–790, 2016; J Sci Comput 67(2):791–794, 2016). In this paper, we have extended the stability analysis to the advection–diffusion equation, demonstrating that the tensor product FR formulation is stable on Cartesian meshes for the advection–diffusion case as well. The analysis in this paper also provides additional insights into the dependence on the VCJH parameter of the diffusion and stability characteristics of these schemes. Several numerical experiments that support the theoretical results are included.