Impact and Importance of Hyperdiffusion on the Spectral Element Method: A Linear Dispersion Analysis

TitleImpact and Importance of Hyperdiffusion on the Spectral Element Method: A Linear Dispersion Analysis
Publication TypeJournal Article
Year of Publication2018
AuthorsUllrich, Paul A., Reynolds Daniel R., Guerra Jorge E., and Taylor Mark A.
JournalJournal of Computational Physics
Date Published06/2018
Abstract / Summary

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.

URLhttp://dx.doi.org/10.1016/j.jcp.2018.06.035
DOI10.1016/j.jcp.2018.06.035
Journal: Journal of Computational Physics
Year of Publication: 2018
Date Published: 06/2018

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.

DOI: 10.1016/j.jcp.2018.06.035
Citation:
Ullrich, PA, DR Reynolds, JE Guerra, and MA Taylor.  2018.  "Impact and Importance of Hyperdiffusion on the Spectral Element Method: A Linear Dispersion Analysis."  Journal of Computational Physics.  https://doi.org/10.1016/j.jcp.2018.06.035.