High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations

Experiments with different discretizations for the shallow‐water equations on a sphere

“…Solid lines and dashed lines indicate positive and negative, respectively. In Figure 8 (top left), the geopotential of REF (1080 × 540, t = 240 s) at day 14 looks similar to the highresolution solution obtained by the German Weather Service (Bernard et al, 2009;their figure 14). The longitudinal and cross-equatorial symmetry of the Rossby wave is well maintained.…”

“…This case has no analytic solution, as do the Rossby-Haurwitz wave (Test 6) and the flow with prescribed initial conditions (Test 7). In Figure 6 (top left), the geopotential of REF (1080 × 540, t = 240 s) at day 15 looks similar to the high-resolution solution obtained by the German Weather Service (Bernard et al, 2009;their figure 12) using a spectral method based on the National Center for Atmospheric Research (NCAR) Spectral Model Transform Shallow Water Model (Jakob et al, 1993). Because the semi-Lagrangian interpolation smooths the fields in REF, the height field is without any spectral ringing or evidence of instability in the vicinity of the mountains.…”

“…In general, we use a grid with fixed resolution and the analysis of meshrefinement techniques discussed by St-Cyr et al (2008) and Weller et al (2009) is not included. We expect that the results presented here will complement recent experiments with finite-volume and discontinuous Galerkin schemes on an icosahedral grid (Läuter et al, 2008;Walko and Avissar, 2008;Bernard et al, 2009;Lee and MacDonald, 2009;Ii and Xiao, 2010), on a cubed-sphere grid Nair, 2009) and on a Yin-Yang grid (Li et al, 2008). This article is organized as follows.…”

From h to p Efficiently: Selecting the Optimal Spectral/hpDiscretisation in Three Dimensions

“…Their ability to handle complex geometries, provide localised refinement in regions of high solution gradient and attain convergence at exponential rates makes them a valuable tool in a wide variety of applications across academia and industry. These originally included problems in fluid dynamics [11], but spectral/ℎ element methods are now used in many other fields such as studies of electromagnetics [6], shallow-water problems [1] and structural dynamics [9].…”

Validating optimisations for chaotic simulations

“…The shallow water model is frequently used to test new techniques for use in weather and climate models [2], [3], [4], [5]. The shallow water model we investigate is a finite difference based model set up to model the wind forced double Munk gyre test case [6], [7] which has features which resemble the Subtropical and Subpolar Gyres of the North Atlantic Surface circulation [8].…”