A corrected Raviart–Thomas Spectral Difference scheme stable for arbitrary order of accuracy on triangular and tetrahedral meshes

Abstract

A theoretical stability estimate is presented for the Raviart–Thomas Spectral Difference scheme for triangular and tetrahedral meshes. It reveals the difficulty of guaranteeing the existence of stable flux points for an arbitrary order of accuracy. To obtain a numerical method stable for any order of accuracy and any space dimension, the present article introduces a new correction term of the flux approximation to cancel the numerical error caused by the Riemann solver exactly. This modification enables the introduction of a new family of so-called Corrected Raviart–Thomas Spectral Difference stable for any polynomial order of approximation. Stability estimates are provided for the L2 norm and a weighted broken Sobolev norm. Numerical experiments in 2D and 3D are conducted for solving linear and non-linear hyperbolic equations with a polynomial order up to p=8 for triangular elements and up to p=5 for tetrahedral elements. They confirm the accuracy, stability, and robustness of the newly introduced numerical scheme. The new method is shown to have a computational cost similar to the Raviart–Thomas Spectral Difference with – for a given mesh – a higher accuracy.