Dispersion analysis of the spectral element method

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. Copyright (c) 2012 Royal Meteorological Society and British Crown Copyright, the Met Office