Abstract
Dimensionally split advection schemes are attractive for atmospheric modelling due to their efficiency and accuracy in each spatial dimension. Accurate long time steps can be achieved without significant cost using the flux‐form semi‐Lagrangian technique. The dimensionally split scheme used in this paper is constructed from the one‐dimensional Piecewise Parabolic Method and extended to two dimensions using COSMIC splitting. The dimensionally split scheme is compared with a genuinely multi‐dimensional, method‐of‐lines scheme which, with implicit time‐stepping, is stable for Courant numbers significantly larger than 1.Two‐dimensional advection test cases on Cartesian planes are proposed which avoid the complexities of a spherical domain or multi‐panel meshes. These are solid‐body rotation, horizontal advection over orography and deformational flow. The test cases use distorted non‐orthogonal meshes either to represent sloping terrain or to mimic the distortions near cubed‐sphere edges.Mesh distortions are expected to accentuate the errors associated with dimension splitting, however the accuracy of the dimensionally split scheme decreases only a little in the presence of mesh distortions. The dimensionally split scheme also loses some accuracy when long time steps are used. The multi‐dimensional scheme is almost entirely insensitive to mesh distortions and asymptotes to second‐order accuracy at high resolution. As is expected for implicit time‐stepping, phase errors occur when using long time steps but the spatially well‐resolved features are advected at the correct speed and the multi‐dimensional scheme is always stable.A naive estimate of computational cost (number of multiplies) reveals that the implicit scheme is the most expensive, particularly for large Courant numbers. If the multi‐dimensional scheme is used instead with explicit time‐stepping, the Courant number is restricted to less than 1, the accuracy is maintained, and the cost becomes similar to the dimensionally split scheme.
Highlights
Many traditional weather and climate models use latitude-longitude meshes, but new models are being developed on quasiuniform meshes in order to better exploit modern computers (e.g. Skamarock and Gassmann, 2011; Staniforth and Thuburn, 2012; Weller et al, 2012; Lauritzen et al, 2014; Katta et al, 2015)
Journal of the Royal Meteorological Society published by John Wiley & Sons Ltd on behalf of the Royal Meteorological Society
We address the issue of conservative, accurate, efficient advection schemes on logically rectangular, non-orthogonal meshes which are stable in the presence of large Courant numbers
Summary
Many traditional weather and climate models use latitude-longitude meshes, but new models are being developed on quasiuniform meshes in order to better exploit modern computers (e.g. Skamarock and Gassmann, 2011; Staniforth and Thuburn, 2012; Weller et al, 2012; Lauritzen et al, 2014; Katta et al, 2015). We address the issue of conservative, accurate, efficient advection schemes on logically rectangular, non-orthogonal meshes which are stable in the presence of large Courant numbers These schemes would be relevant for cubed-sphere meshes and for terrain-following meshes. These are: the solid-body rotation test case of Leonard et al (1996), modified to use a mesh (or co-ordinate system) with distortions similar to a cubed-sphere (section 3.1); the horizontal advection test case over orography (Schar et al, 2002), examining sensitivity to time step, resolution and mountain height, all on the basic terrain following mesh without smoothing of terrain following layers (section 3.2); and a modification of the deformational flow test case of Lauritzen et al (2012) for a periodic rectangular plane (section 3.3). ∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z (Pudykiewicz and Staniforth, 1984) such that cd ≤ 1
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