The Spectral Difference Raviart-Thomas method for two and three-dimensional elements and its connection with the Flux Reconstruction formulation

TL;DR

This paper develops the Spectral Difference Raviart-Thomas (SDRT) method for 2D and 3D elements, establishes its connection with Flux Reconstruction schemes, analyzes stability and errors, and evaluates performance on GPU architectures.

Contribution

It introduces the SDRT formulation for tensor-product and simplex elements, links it with FR methods, and provides a comprehensive analysis of stability, dissipation, dispersion, and computational performance.

Findings

SDRT has enhanced temporal stability compared to FR-DG.

SDRT exhibits larger dissipation and dispersion errors than FR-DG.

Numerical experiments reveal non-linear instability of SDRT with simplex elements.

Abstract

The purpose of this work is to describe in detail the development of the Spectral Difference Raviart-Thomas (SDRT) formulation for two and three-dimensional tensor-product elements and simplexes. Through the process, the authors establish the equivalence between the SDRT method and the Flux-Reconstruction (FR) approach under the assumption of the linearity of the flux and the mesh uniformity. Such a connection allows to build a new family of FR schemes for two and three-dimensional simplexes and also to recover the well-known FR-SD method with tensor-product elements. In addition, a thorough analysis of the numerical dissipation and dispersion of both aforementioned schemes and the nodal Discontinuous Galerkin FR (FR-DG) method with two and three-dimensional elements is proposed through the use of the combined-mode Fourier approach. SDRT is shown to possess an enhanced temporal linear stability regarding the FR-DG. On the contrary, SDRT displays larger dissipation and dispersion errors with respect to FR-DG. Finally, the study is concluded with a set of numerical experiments, the linear advection-diffusion problem, the Isentropic Euler Vortex and the Taylor-Green Vortex (TGV). The latter test case shows that SDRT schemes present a non-linear unstable behavior with simplex elements and certain polynomial degrees. For the sake of completeness, the matrix form of the SDRT method is developed and the computational performance of SDRT with respect to FR schemes is evaluated using GPU architectures.