Dispersive and Dissipative Behavior of the Spectral Element Method

Ainsworth, Mark; Wajid, Hafiz Abdul

doi:10.1137/080724976

Cited by 91 publications

(76 citation statements)

References 23 publications

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“…However this is not the focus of the present study, but just a natural consequence. Following submission of the present analysis for publication, we became aware of the related analysis of Ainsworth and Wajid (2009), which complements the present one. Both analyses examine one-dimensional linear wave dispersion for SEM discretisations, and both provide valuable insight, but the approaches taken, and the contexts for the analyses, are somewhat different.…”

Section: Introductionmentioning

confidence: 99%

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Dispersion analysis of the spectral element method

Melvin

Staniforth

Thuburn

2012

Quart J Royal Meteoro Soc

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† The contributions of these authors were written in the course of their employment at the Met Office, UK, and are published with the permission of the Controller of HMSO and the Queen's Printer for Scotland.The spectral element method (SEM) has (with exact time integration) the desirable attribute of locally and globally conserving mass, energy and potential vorticity. It also scales well on massively parallel computers. Another desirable attribute of a numerical method for an atmospheric dynamical core is that it should have good numerical dispersion properties in order to accurately represent wave propagation and adjustment processes. Application of the SEM to the one-way wave equation is analysed to provide insight into its dispersion properties as a function of spectral order. For the lowest-order spectral truncation (linear) the SEM discretisation is formally equivalent to centred second-order finite differences on an Arakawa A grid. It consequently shares its poor dispersion properties, including energy propagation in the wrong direction for the short-wavelength half of the spectrum. Increasing the spectral truncation of the SEM to quadratic improves its dispersion properties for the long-wavelength part of the spectrum, but the problem of energy propagation in the wrong direction for the short-wavelength part remains. Further increasing the order of the spectral truncation not only fails to address the poor energy propagation at small scales, but also introduces new problems, including gaps in the spectrum of frequencies that can be represented, and localisation of eigenmode structures near element boundaries. Numerical integrations confirm that these SEM dispersion properties lead to reversed group velocities and to grid imprinting at spectral element boundaries.

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Section: Introductionmentioning

confidence: 99%

“…(1)-(2) and the equivalent (3)-since this mimics what is done in this context, and it is therefore the approach adopted herein. However, Ainsworth and Wajid (2009) examine the SEM discretisation of the second-order wave equation…”

Section: Introductionmentioning

confidence: 99%

Dispersion analysis of the spectral element method

Melvin

Staniforth

Thuburn

2012

Quart J Royal Meteoro Soc

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“…Many methods have been proposed for studying the dispersive properties of various numerical schemes (e.g., finite difference method (FDM), FEM, and SEM) for many problems, such as wave propagation, linear convection diffusion, and Helmholtz equations. Some methods measure dispersion, such as eigenvalue [13][14][15][16][17][18], wavenumber [12,[19][20][21][22], angular frequency [23][24][25], and error derivation [26][27][28][29] methods. However, such dispersion analysis considers only spatial discretization.…”

Section: Introductionmentioning

confidence: 99%

Space-time spectral element method solution for the acoustic wave equation and its dispersion analysis

Geng

Qin

Zhang

et al. 2017

Acoust. Sci. & Tech.

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This paper presents the solution for the continuous space-time spectral element method (CSTSEM) based on Chebyshev polynomials for the acoustic wave equation. Acoustic wave propagation in various dimensions is simulated using quadrilateral, hexahedral, and tesseract elements. The convergence is studied for 1+1-dimensional wave propagation. The extended 2+1-and 3+1-dimensional wave equations are also numerically solved by CSTSEM and the dispersion characteristics are investigated. A fixed-! (! is the angular frequency) method is proposed for computing the dispersion of space-time coupled methods. CSTSEM is verified as a simple, practical isotropic algorithm with low dispersion and high accuracy.

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“…Here, our goal is to evaluate and improve the multi-resolution capabilities of the SEM formulation from the High-Order Method Modeling Environment (HOMME), recently adopted as the default dynamical core by the Community Atmosphere Model (CAM) (Dennis et al, 2012). HOMME uses a locally conservative/mimetic formulation from Taylor and Fournier (2010) and relies on a constant-coefficient hyperviscosity term to both dissipate energy near the grid scale and to damp grid scale modes with spurious propagation (Ainsworth and Wajid, 2009). This hyperviscosity operator is not suitable for variable-resolution grids, and thus we consider two resolution-aware extensions.…”

Section: Introductionmentioning

confidence: 99%

The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity

et al. 2014

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Abstract. We evaluate the performance of the Community Atmosphere Model's (CAM) spectral element method on variable-resolution grids using the shallow-water equations in spherical geometry. We configure the method as it is used in CAM, with dissipation of grid scale variance, implemented using hyperviscosity. Hyperviscosity is highly scale selective and grid independent, but does require a resolutiondependent coefficient. For the spectral element method with variable-resolution grids and highly distorted elements, we obtain the best results if we introduce a tensor-based hyperviscosity with tensor coefficients tied to the eigenvalues of the local element metric tensor. The tensor hyperviscosity is constructed so that, for regions of uniform resolution, it matches the traditional constant-coefficient hyperviscosity. With the tensor hyperviscosity, the large-scale solution is almost completely unaffected by the presence of grid refinement. This later point is important for climate applications in which long term climatological averages can be imprinted by stationary inhomogeneities in the truncation error. We also evaluate the robustness of the approach with respect to grid quality by considering unstructured conforming quadrilateral grids generated with a well-known grid-generating toolkit and grids generated by SQuadGen, a new open source alternative which produces lower valence nodes.

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Dispersive and Dissipative Behavior of the Spectral Element Method

Cited by 91 publications

References 23 publications

Dispersion analysis of the spectral element method

Dispersion analysis of the spectral element method

Space-time spectral element method solution for the acoustic wave equation and its dispersion analysis

The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity

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