Comparison of dimensionally split and multi‐dimensional atmospheric transport schemes for long time steps

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-oflines 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 multidimensional 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.

A Holistic Algorithmic Approach to Improving Accuracy, Robustness, and Computational Efficiency for Atmospheric Dynamics

“…This creates a 2nd-order error term in time when handled via an alternating Strang splitting, the constant of which is quite small in orthogonal coordinates. In nonorthogonal coordinates, there are other dimensionally split approaches that also result in low splitting error (e.g., [16,2]), and often, an unsplit scheme is warranted [23]. A more formal description of the FV method is given in Appendix B.…”

Semi-implicit second order schemes for numerical solution of level set advection equation on Cartesian grids

“…where S denotes an amplification factor, the von Neumann stability analysis is realized by searching for conditions under which one has |S| ≤ 1 for all h ∈ (−π, π) . Using (5) in (3) one obtains…”

“…We consider the linear advection equation on Cartesian grids also as a starting point for a study of more complex equations like a nonlinear advection equation for a motion in normal direction [39,35,12,30,14] and computations on unstructured grids [12,9,17]. We are interested here in Eulerian type of numerical schemes of a finite difference form when a stencil of the scheme does not move in time like in Lagrangian type of numerical schemes [11,5]. Furthermore we restrict ourselves to the schemes using an implicit or a semi-implicit time discretization with a purpose of favorable stability properties when compared to the schemes of Eulerian type using a fully explicit time discretization.…”

A high‐order WENO‐limited finite‐volume algorithm for atmospheric flow using the ADER‐differential transform time discretization

“…Dimensional splitting for finite-volume methods is desirable computationally because it typically reduces the computational complexity of the scheme by a factor of N (D−1) , where N is the order of accuracy of the scheme, and D is the number of spatial dimensions. Most splittings will suffice in an orthogonal coordinate system as shown in section 3.2.1, but in nonorthogonal coordinates, a straightforward Strang splitting gives very poor accuracy Norman and Nair (2018); Chen et al (2017), though coupling dimensions through a multistage time integrator is still accurate in nonorthogonal coordinates Katta et al (2015). However, for ADER-DT, there is no existing implementation that uses stages.…”

“…Most splittings will suffice in an orthogonal coordinate system as shown in section 3.2.1, but in nonorthogonal coordinates, a straightforward Strang splitting gives very poor accuracy Norman and Nair (2018); Chen et al . (2017), though coupling dimensions through a multistage time integrator is still accurate in nonorthogonal coordinates Katta et al . (2015).…”