On the Stability of the Flux Reconstruction Schemes on Quadrilateral Elements for the Linear Advection Equation

The flux reconstruction (FR) approach to high-order methods has proved to be a promising alternative to traditional discontinuous Galerkin (DG) schemes since they facilitate the adoption of explicit time-stepping methods suitable for parallel architectures like GPUs. The FR approach provides a parameterized family of schemes through which various classical schemes like nodal-DG and spectral difference methods can be recovered. Further, the parameters can be varied to obtain schemes with a maximum stable time-step, or minimum dispersion or dissipation errors etc., providing us a single powerful framework unifying high-order discontinuous Finite Element Methods. There have been various studies on the accuracy and the stability of these schemes and in particular, a subset of the FR schemes known as ESFR or VCJH schemes have been shown to be stable in 1D and on simplex elements in 2D and 3D for the linear advection as well as the advection–diffusion equations. However, the stability of the FR schemes on tensor product quadrilateral elements has remained an open question. Although it is the most natural extension of the 1D FR approach, it has posed a significant challenge, especially for general quadrilateral elements. In this paper, we investigate the stability of the VCJH-type FR schemes for linear advection on Cartesian quadrilateral meshes and show that the schemes could become unstable under certain conditions. However, we find that the VCJH scheme recovering the DG method is stable on all Cartesian meshes. Although we restrict ourselves to Cartesian meshes in order to circumvent the algebraic complexity posed by the variation of the Jacobian matrix inside general tensor-product quadrilateral elements, our analysis offers significant insight into the possible origins of instability in the FR approach on general quadrilaterals.