A stable high-order Spectral Difference method for hyperbolic conservation laws on triangular elements

Numerical schemes using piecewise polynomial approximation are very popular for high order discretization of conservation laws. While the most widely used numerical scheme under this paradigm appears to be the Discontinuous Galerkin method, the Spectral Difference scheme has often been found attractive as well, because of its simplicity of formulation and implementation. However, recently it has been shown that the scheme is not linearly stable on triangles. In this paper we present an alternate formulation of the scheme, featuring a new flux interpolation technique using Raviart–Thomas spaces, which proves stable under a similar linear analysis in which the standard scheme failed. We demonstrate viability of the concept by showing linear stability both in the semi-discrete sense and for time stepping schemes of the SSP Runge–Kutta type. Furthermore, we present convergence studies, as well as case studies in compressible flow simulation using the Euler equations.