By using CSL3 multimoment interpolation, a piecewise cubic polynomial for spatial reconstruction can be obtained with four multimoment constraint conditions consisting of two point values at cell boundaries, one volume-integrated average and one slope parameter at the cell center. The resulting multimoment finite-volume scheme is of fourth-order accuracy. A non-oscillatory scheme can be derived by designing the proper formula to calculate the slope parameter at the cell center. A new strategy was recently proposed, using the Weighted Essentially Non-Oscillatory (WENO) concept to determine the slope parameter. Using a WENO-type limiter, the multimoment reconstruction can effectively remove nonphysical oscillations while keeping fourth-order accuracy in smooth regions. In this study, a WENO-type slope limiter is proposed and implemented in our multimoment finite-volume global transport model based on the cubed-sphere grid. The widely used benchmark tests, including both solid rotation and complicated deformational advection cases, are checked to verify the performance of the proposed global transport model. Numerical results reveal that a WENO-type slope limiter can greatly improve the accuracy of the multimoment finite-volume model compared with the former Total Variation Diminishing (TVD)-type limiter. Furthermore, the proposed limiter is constructed over a compact stencil of only three adjacent cells. Without any user-defined or problem-dependent parameters, the present model is very promising for practical applications. KEYWORDS cubed-sphere grid; global model; multimoment scheme; slope limiter; transport model; WENO
INTRODUCTIONDue to polar problems on the latitude-longitude grid, global computational meshes with a quasi-uniform grid spacing and free of polar problems, such as the cubed-sphere grid (Sadourny, 1972), icosahedral grid (Sadourny et al., 1968;Williamson, 1968), and yin-yang grid (Kageyama and Sato, 2004), are becoming popular in constructing global models, especially for those aiming at very high-resolution simulations of atmospheric dynamics. One of the key issues in implementing research and operational models on these quasi-uniform grids is the relatively complex grid structure, which causes essential difficulties in constructing high-order models and introduces extra grid-imprinting errors. In the past decade, some accurate global models have been proposed on these grids through the application of so-called "local" schemes, such as the discontinuous Galerkin scheme (Nair et al., 2005a;2005b), the spectral element scheme (Dennis et al., 2012), the multimoment scheme (Chen et al., 2014) and so on. These schemes are accomplished based on compact stencils (usually within a single computational element) and are very convenient to implement in spherical geometry to develop high-order models. Furthermore, these schemes are computationally intensive, so they are highly scalable on massively parallel clusters. A comprehensive review on the applications of quasi-uniform grids can be found in the work of...
An oscillation‐less multimoment global transport model was proposed by Tang et al. in 2018 by introducing a slope limiter based on the weighted essentially non‐oscillatory (WENO) concept. The spurious oscillations around the discontinuities and strong gradients can be effectively removed and the model preserves the fourth‐order accuracy in spherical geometry for smooth solutions. However, the WENO limiter does not strictly guarantee the non‐negativity of numerical solution and the resulting model will violate the physical principle due to the existence of the nonphysical negative undershoots. As the numerical fluxes, which determine the time tendency of the volume‐integrated average, are evaluated by the point values (PVs) defined along the cell boundaries in the multimoment finite‐volume schemes, the non‐negativity preserving model can be accomplished by modifying those PVs to implement the corrections on the numerical fluxes proposed in the positive definite flux‐corrected transport (FCT) method by Smolarkiewicz in 1989. In this study, a non‐negativity correction algorithm is proposed for a global transport model using the MM‐FVM_WENO scheme and verified by simulating the widely used benchmark tests.
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 transport model on a cubed‐sphere grid is presented using a multimoment finite‐volume (MMFV) method. The three‐point fourth‐order MMFV method with boundary gradient switching projection is utilized to ensure nonoscillatory numerical solutions in the horizontal, while the simple and computationally efficient semi‐Lagrangian piecewise rational method is used to allow a large time step in the vertical. By means of horizontal and vertical separation, a Strang‐type splitting technique is adopted to advance the numerical solutions in time. Non‐negative corrections are made to achieve positive‐definite preservation exactly. The resulting transport model is inherently conservative, high‐accuracy, and positive‐definite, and allows a large Courant number no matter how dense the vertical grid is. Various benchmark test cases, including three representative advection tests and an idealized terminator chemistry test, are conducted to validate the performance of the model presented. The numerical results indicate that the model is quite competitive in comparison with the existing models in the literature and looks very promising in real atmospheric simulations.
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