Impact and importance of hyperdiffusion on the spectral element method: A linear dispersion analysis

Highlights: • Hyperdiffusion for SEM improves the dispersive properties of discrete wave modes. • The KG53 scheme with time-split diffusion is the most efficient method investigated. • With hyperdiffusion, SEM is physically consistent for waves longer than $3\Delta x$. • Scalar and/or divergence damping is effective at eliminating the spectral gap in 2D. • Analysis routines for SEM with hyperdiffusion have been developed for public use. The spectral element method (SEM) is a mimetic finite element method with several properties that make it a desirable choice for numerical modeling. Although the linear dispersion properties of this method have been analyzed extensively for the case of the 1D inviscid advection equation, practical implementations of the SEM frequently employ hyperdiffusion for stabilization. As argued in this paper, hyperdiffusion has a pronounced impact on the accuracy of the discrete wave modes and the dispersive properties of the SEM. When applied with an appropriately large coefficient, hyperdiffusion is effective at removing the spectral gap and improving the stability of the 1D advection equation. This study also considers the SEM as applied to the 2D linearized shallow-water equations, where hyperdiffusion in the form of scalar diffusion, divergence damping, and vorticity damping are analyzed. To the extent possible, guidance on the choice of hyperdiffusion coefficients is provided. A brief discussion of the comparative impact of local element filtering is included.