A Positivity-preserving Conservative Semi-Lagrangian Multi-moment Global Transport Model on the Cubed Sphere

Abstract

A positivity-preserving conservative semi-Lagrangian transport model by multi-moment finite volume method has been developed on the cubed-sphere grid. Two kinds of moments (i.e., point values (PV moment) at cell interfaces and volume integrated average (VIA moment) value) are defined within a single cell. The PV moment is updated by a conventional semi-Lagrangian method, while the VIA moment is cast by the flux form formulation to assure the exact numerical conservation. Different from the spatial approximation used in the CSL2 (conservative semi-Lagrangian scheme with second order polynomial function) scheme, a monotonic rational function which can effectively remove non-physical oscillations is reconstructed within a single cell by the PV moments and VIA moment. To achieve exactly positive-definite preserving, two kinds of corrections are made on the original conservative semi-Lagrangian with rational function (CSLR) scheme. The resulting scheme is inherently conservative, non-negative, and allows a Courant number larger than one. Moreover, the spatial reconstruction can be performed within a single cell, which is very efficient and economical for practical implementation. In addition, a dimension-splitting approach coupled with multi-moment finite volume scheme is adopted on cubed-sphere geometry, which benefitsthe implementation of the 1D CSLR solver with large Courant number. The proposed model is evaluated by several widely used benchmark tests on cubed-sphere geometry. Numerical results show that the proposed transport model can effectively remove nonphysical oscillations and preserve the numerical non-negativity, and it has the potential to transport the tracers accurately in a real atmospheric model.

A three‐dimensional positivity‐preserving and conservative multimoment finite‐volume transport model on a cubed‐sphere grid

“…In order to preserve positivity, we make the modifications introduced in our previous work (Tang et al, 2021). Equation () used to calculate the interface value may produce negative values; an easy and effective modification of PV moment for the positive‐definite case can be implemented by $$$$ {\overline{{}^P\phi}}_{k+\frac{1}{2}}=\max \left(0,{\overline{{}^P\phi}}_{k+\frac{1}{2}}\right).$$…”

“…In order to preserve positivity, we make the modifications introduced in our previous work (Tang et al, 2021).…”

“…The resulting model can maintain high-order accuracy in the horizontal direction and permits a large Courant number in the vertical direction, which is very computationally efficient and beneficial for practical application. Furthermore, two non-negativity corrections in our previous work (Li et al, 2020;Tang et al, 2021) are implemented in the transport model to ensure a positive numerical solution. As a whole, the novelties of this study are as follows: (1) development of a conservative, nonoscillatory, and positive-definite transport model, which is suitable for real weather and climate simulations; (2) extension of the MCV3-BGS algorithm (Deng et al, 2017) with non-negativity correction (Li et al 2020) into the cubed-sphere grid in the horizontal; (3) combination of the PRM scheme (Xiao and Peng, 2004) with modified positive-definite correction (Tang et al, 2021) with a fourth-order non-negativity MCV3-BGS algorithm to assure the positivity of the 3D transport model.…”

“…As a whole, the novelties of this study are as follows: (1) development of a conservative, nonoscillatory, and positive‐definite transport model, which is suitable for real weather and climate simulations; (2) extension of the MCV3‐BGS algorithm (Deng et al, 2017) with non‐negativity correction (Li et al . 2020) into the cubed‐sphere grid in the horizontal; (3) combination of the PRM scheme (Xiao and Peng, 2004) with modified positive‐definite correction (Tang et al, 2021) with a fourth‐order non‐negativity MCV3‐BGS algorithm to assure the positivity of the 3D transport model.…”

High-Order Semi-Lagrangian Schemes for the Transport Equation on Icosahedron Spherical Grids

“…The traditional latitude-longitude grid is easy for application but has singularities at the two poles [4]. Moreover, its non-uniform grid system would also seriously affect computational efficiency [5]. To address these issues, quasiuniform grid systems without singularities or with weak singularities, such as Yin-Yang overset grids, block-structured quadrilateral meshes based on the cubed sphere, triangle or their dual meshes based on the regular icosahedron sphere, are becoming more and more popular in developing global transport models [6,7].…”