A global finite-element shallow-water model supporting continuous and discontinuous elements

Abstract

Abstract. This paper presents a novel nodal finite-element method for either continuous and discontinuous elements, as applied to the 2-D shallow-water equations on the cubed sphere. The cornerstone of this method is the construction of a robust derivative operator that can be applied to compute discrete derivatives even over a discontinuous function space. A key advantage of the robust derivative is that it can be applied to partial differential equations in either a conservative or a non-conservative form. However, it is also shown that discontinuous penalization is required to recover the correct order of accuracy for discontinuous elements. Two versions with discontinuous elements are examined, using either the g 1 and g 2 flux correction function for distribution of boundary fluxes and penalty across nodal points. Scalar and vector hyperviscosity (HV) operators valid for both continuous and discontinuous elements are also derived for stabilization and removal of grid-scale noise. This method is validated using four standard shallow-water test cases, including geostrophically balanced flow, a mountain-induced Rossby wave train, the Rossby-Haurwitz wave and a barotropic instability. The results show that although the discontinuous basis requires a smaller time step size than that required for continuous elements, the method exhibits better stability and accuracy properties in the absence of hyperviscosity.

DCMIP2016: a review of non-hydrostatic dynamical core design and intercomparison of participating models

“…More information on this choice of grid can be found in Ullrich (2014a). On the equiangular cubed-sphere grid, coordinates are given as (α, β, p), with central angles α, β ∈ [− π 4 , π 4 ] and panel index p. The structure of this grid supports refinement through stretching (Schmidt, 1977;Harris et al, 2016) or nesting (Harris and Lin, 2013).…”

“…The horizontal placement of variables impacts a number of properties of the numerical method, including how energy and enstrophy conservation is managed, any computational modes that might arise due to differencing, dispersion properties, and the maximum stable time-step size for explicit time-stepping schemes (Randall, 1994;Ullrich, 2014b). The original four Arakawa grids (Arakawa and Lamb, 1977), denoted with letters A through D, were initially designed for rectilinear meshes but were later adapted for a variety of unstructured grids.…”

“…In both ACME-A and Tempest, scalar hyperviscosity is employed for ρ, θ, and tracer variables via repeated application of a scalar Laplacian (Dennis et al, 2012;Ullrich, 2014a). Vector hyperviscosity is also applied by decomposing the horizontal vector Laplacian into divergence damping and vorticity damping terms via the vector identity…”

“…3.4) optimized with spring dynamics using the method of Tomita et al (2002). A terrain-following height coordinate system is used in the vertical (Tomita and Satoh, 2004) with Lorenz staggering.…”

“…The Tempest model (Ullrich, 2014a;Guerra and Ullrich, 2016) is an experimental test bed for high-performance numerical methods that solves the non-hydrostatic Euler equations on a cubed-sphere grid (Sect. 3.2) using a horizontally co-located spectral element discretization.…”