1. Introduction
Because of inherent conservative properties and geometric flexibility, finite-volume-based (FV) discretization techniques are becoming popular for new generation global atmospheric models. The cubed-sphere grid system (Sadourny 1972; Ronchi et al. 1996) provides quasi-uniform grid structures (control volumes) for atmospheric modeling, which is also an ideal system for FV horizontal discretization. The cubed-sphere grid system is free of polar singularities and the control volumes (grid cells) are logically rectangular leading to efficient parallel implementation (Yang and Cai 2011). In recent years, several new models have been developed that exploit computationally attractive features associated with the FV discretization and cubed-sphere geometry (Putman and Lin 2007; Cheruvu et al. 2007; Chen and Xiao 2008; Ullrich et al. 2010).
The cubed-sphere consists of six identical spherical surfaces defined by local coordinate systems that are discontinuous at the edges and corners. Therefore, a major difficulty in adopting the cubed-sphere geometry arises from the “handling” of the edges, where a special treatment is required. As the order of the discretization increases, the issue becomes more complex.
To predict the cell averages at the new time level, FV methods require a reconstruction procedure for fluxes at the cell edges from the known cell averages. This involves a computational halo region (stencil) encompassing several grid cells. A fully two-dimensional (2D) FV approach requires ghost cell creation at the cubed-sphere corner. However, a dimension-by-dimension approach employing two 1D reconstructions along the coordinate directions greatly simplifies the problem. A major concern with the dimension-by-dimension approach, the resulting FV scheme suffers from reduction in formal order of accuracy, and this issue might be more severe in nonorthogonal curvilinear grid such as cubed-sphere grid. This motivates us to compare the performance of 1D and 2D reconstruction high-order FV schemes for a variety of benchmark tests on the cubed sphere.
We consider a high-order FV discretization based on the so-called central-upwind finite-volume (CUFV) method introduced by Kurganov and Levy (2000) and Kurganov and Petrova (2001). The CUFV scheme is a semidiscretized method combining the attractive properties of the classical upwind and central FV methods. Its features include easy A-grid (unstaggered) implementation with simple Riemann solvers (numerical flux). Because of its semidiscretized (spatially discretized) formulation, the time integration can be performed by explicit multistage Runge–Kutta (RK) solvers resulting in high-order temporal accuracy and increased Courant–Friedrichs–Lewy (CFL) stability limit. A recent application of CUFV method for ocean and atmospheric modeling can be found in (Adamy et al. 2010; Nair and Katta 2013). For the present work, we consider two high-order spatial discretizations (reconstructions). The dimension-by-dimension version of the FV scheme is based on the fifth-order Weighted Essentially Nonoscillatory (WENO5) method (Liu et al. 1994; Shu 1997). For multidimensional application, high-order 2D WENO schemes are computationally prohibitive and rarely used for practical purpose. Therefore, we consider a fully 2D fourth-order FV discretization as given in Kurganov and Liu (2012). Our main focus here is to evaluate the dimension-by-dimension WENO5 reconstructions in a CUFV framework for linear transport problem on a nonorthogonal curvilinear cubed-sphere grid. The performance of WENO5 scheme is compared with a CUFV scheme based on 2D reconstructions as well as various other high-order FV schemes developed on the cubed sphere. In addition, we discuss strictly positivity-preserving filters for both CUFV schemes.
The paper is organized as follows. Section 2 describes CUFV schemes based on 1D and 2D reconstructions and its implementation on cubed sphere. In section 3, time integration schemes and positivity-preserving filters are discussed. Numerical experiments are described in section 4, followed by summary and conclusions in section 5.
2. CUFV formulation
a. 2D linear transport on cubed sphere




















Schematic showing a cubed sphere (a) with rectangular FV cells, total
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-13-00176.1







b. CUFV schemes
A large class of FV methods for solving hyperbolic conservation laws are based on high-order extensions of the Godunov scheme (Godunov 1959), collectively known as the Godunov-type schemes (Toro 1999). These schemes essentially have three basic steps in the solution process: reconstruction, evolution, and projection. In reconstruction step, piecewise polynomials are reconstructed over the grid cells spanning the domain from the known cell averages (piecewise constant data) at the previous time level (van Leer 1974; Colella and Woodward 1984). In evolution step, the piecewise polynomials are advanced in time, following the underlying conservation law. At the final projection step, new cell averages are computed on each cell by projecting the evolved polynomials onto cell averages. Such Godunov-type schemes are broadly classified into upwind and central schemes. The CUFV combines these two methods resulting in a class of semidiscrete (continuous in time) scheme, which are relatively simple and are easy to implement in various applications. Its novel features include high-order accuracy, use of simple numerical flux, and can be implemented in a nonstaggered grid system when used for a system of equations. These make CUFV computationally attractive for complex domain such as the cubed-sphere considered here. Detailed discussion of CUFV schemes including mathematical derivations, properties, and various practical applications can be found in a series of papers (see Kurganov and Levy 2000; Kurganov and Petrova 2001; Kurganov and Liu 2012).























1) Dimension-by-dimension fifth-order WENO reconstructions
The dimension-by-dimension case combines two sweeps of 1D polynomial functions along the coordinate direction and is subject to the conservation constraint (5). The WENO schemes are known to be robust for solving conservation laws. A comprehensive review for WENO scheme is given in Shu (1997). One can rigorously derive a fifth-order accurate fully 2D WENO scheme using a
In Fig. 2, a 2D stencil used for the WENO5 is schematically shown with cell centers in the west–east and south–north directions. Flux evaluation for the WENO5 scheme is required only at four cell walls as indicated in Fig. 1b, making the computational procedure relatively simple. A typical WENO reconstruction process involves a main computational stencil and several substencils within. The basic idea of the WENO method is to use a convex combination of reconstructions from all the stencils and employ nonlinear weights to achieve highest possible order of accuracy in smooth regions. The WENO scheme uses a convex combination of nonlinear weights
Schematic of the 2D stencil required for KL scheme where 13 cells are used for reconstructing the fluxes along the cell boundaries (green lines) on the central cell. Black boxes indicate the two 1D stencils for WENO5 scheme, along the west–east and south–north directions, excluding the corner cells. A total of nine cells are required for WENO5 reconstructions.
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-13-00176.1












2) Fully 2D reconstructions














3) Treatment at the cubed-sphere edges
High-order FV schemes require a wider computational stencil involving several cells. Because of the coordinate discontinuity at the edges of the cubed-sphere face, creation of such stencils is a challenging task for the cubed-sphere grid system. Each face of the cubed sphere has logically rectangular cells, however, by the virtue of equiangular (central) projection this further simplifies (i.e., in the computational domain
Horizontal extensions of the cubed-sphere grid points at the edges to form halo regions (cells) required for the CUFV computational stencils. For any two adjoining panels, the extended grid points are exactly located along a great-circle arc joining the grid points from the other panel in the vertical direction, which are shown as dashed lines. 1D interpolations are performed along these lines using the cell averages at regular grid points (source points) to find the cell averages for the extended grid points (target points).
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-13-00176.1
To illustrate the 1D interpolation process, we consider two lateral adjoining panels as shown in Fig. 3, where the cell centers are marked as red and blue points, with known spherical coordinates. Let
We use the cubic-Lagrange interpolation along the dotted lines to compute cell averages at the target points (on halo cells)
Handling of the cubed-sphere edges for fully 2D scheme can be performed in a more sophisticated way for better accuracy, but at a higher computational cost. The multimoment FV scheme by Chen and Xiao (2008) identifies one layer of the target cells on the halo zone as described above. To find the ghost-cell values, a 2D interpolation is performed using the readily available local moments. This method seems to be very accurate but only suitable for multimoment FV schemes. Ullrich et al. (2010) proposed novel high-order FV schemes on the cubed-sphere, where the ghost-cell averages are obtained by using the Gauss quadrature over the target cell, which involves sampling the values at the quadrature points by the local reconstruction polynomials. However, for simplicity we do not employ this method for the KL scheme, rather we use the 1D approach described above. Using the common 1D interpolation procedure for both the WENO5 and KL schemes facilitates a closer comparison.
3. Time integrations and positivity filters
a. Time integration scheme: SSP-RK (5,4)




b. Positivity-preserving filters
The WENO schemes can control spurious oscillations in the solution to a great extent, nevertheless, there is no guarantee that it will always keep the numerical solution within the legitimate (physical) bounds. The numerical solution with WENO schemes may still have small amplitude oscillations, in other words, these schemes are only “essentially” nonoscillatory, but not strictly positivity preserving. Another issue is that the final semidiscrete FV equation (8) itself may be a source for tiny spurious negative numbers due to numerical precision errors. This is because on the right side of (8), time tendencies are computed as differences of fluxes through the cell walls, when the values of the fluxes are very close, the net result may have a negative sign (with very small magnitude). For many atmospheric tracers such as humidity and mixing ratios, the global maximum and minimum values are known in advance, moreover, for which negative values are not acceptable. To address this issue we implement optional positivity-preserving filters to the CUFV schemes.
First, we discuss a bound-preserving (BP) conservative filter, which is particularly useful when the global minimum and maximum value of the solution is known in advance. In the present work we implement the BP filter for the schemes considered. The BP filter relies on local reconstruction polynomial, and it is computationally inexpensive. The BP filter is based on the Liu and Tadmor (1998) limiter. Recently, Zhang and Shu (2010) extended this for high-order discontinuous Galerkin (DG) schemes, and Zhang and Nair (2012) implemented the BP filter for a DG transport scheme on the cubed sphere. We apply this filter for both WENO5 and KL reconstruction polynomials.












A scheme is considered to be positive definite, if it does not introduce any negative values in the computed solution from nonnegative initial values. However, because of arithmetic precision errors as mentioned above, the solutions with very small magnitude might still have negative signs. A positivity-preserving (or sign preserving) (PP) filter may be applied at the final stage of computation to completely eliminate unacceptable negative solution. To ensure the positivity of the solution, we employ the PP filter based on an upstream renormalization approach developed by Smolarkiewicz (1989). For oscillations with small amplitude this filter is very robust, and we apply the PP filter as the finalization process for CUFV combined with BP filter. The PP filter is local, computationally cheap, and easy to implement. Recently, Blossey and Durran (2008) implemented the PP filter for their FV schemes, this is in fact, a special case of the flux-corrected transport (FCT) algorithm (Durran 1999). The details of the PP algorithm can be found in Smolarkiewicz (1989). Note that the BP filter is only applicable when the global extrema M and m are known, and it is considered to be a limitation of this approach (Zhang and Nair 2012).
4. Numerical experiments






a. Solid-body rotation tests






















First, we demonstrate the effect of BP and PP filters with the cosine-bell advection test. For this experiment the WENO5 scheme was selected on a
Results of the cosine-bell advection test on the cubed sphere after one revolution (12 days) with the WENO5 scheme. The wind field is oriented along the northeast direction (
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-13-00176.1
To compare the results with other high-order FV models, we conducted additional experiments for the cosine-bell test. At a resolution
Time traces of the normalized errors
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-13-00176.1
This experiment is repeated for a lower grid resolution
Figure 6 shows the convergence of normalized errors (
Convergence results with the solid-body rotation of a Gaussian hill for the WENO5 and KL schemes. The normalized errors (a)
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-13-00176.1
b. Deformational flow test: Moving vortices













The cubed-sphere resolution is chosen to be
Numerical solution with the WENO5 scheme for the moving vortices test. (a) Initial vortex field, (b) solution at halftime (6 days), and (c) solution at after full evolution (12 days). The vortices move along the northeast direction
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-13-00176.1
c. Deformational flow test: Slotted cylinders
To further validate the CUFV schemes on the sphere, we use a challenging benchmark deformational flow test case recently developed by Nair and Lauritzen (2010). We are particularly interested in nonsmooth (twin slotted cylinder) initial conditions. The initial distributions are deformed into thin filaments halfway through the simulation while they are being transported along the zonal direction by the solid-body component of the flow.








Numerical solution for the deformational flow test on a cubed sphere with mesh 90
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-13-00176.1




The same test case can be used for convergence studies, if the slotted cylinders are replaced by two symmetrically located Gaussian hills in (19), as discussed in Nair and Lauritzen (2010). Recently, this test case has been considered in Lauritzen et al. (2012) for comparing various advection schemes. The initial smooth fields (
Figure 8 shows the results of the deformational flow tests with the WENO5 scheme in Figs. 8b and 8d, respectively, at halftime (
Figure 9 shows the convergence of the normalized errors with smooth deformational flow involving double-Gaussian fields. Clearly both WENO5 and KL show more than second-order convergence for the complex flow fields, and the results are comparable to the CSLAM scheme as shown in Lauritzen et al. (2012). The semi-Lagrangian scheme with reduced dependence (flux based) on grid geometry shows a better convergence rate for this test as shown in Erath and Nair (2014). A degradation in the convergence may be due to the fact that both schemes rely on a quadratic interpolation method at the corner (halo) cells of the cubed sphere. A rigorous approach would be employing the compact Hermit interpolation recently introduced by Croisille (2013) or interpolation with localized radial basis functions at the cubed-sphere corners. However, we do not consider these advanced methods for the present study.
Convergence for the deformational flow with double-Gaussian fields for the normalized errors (a)
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-13-00176.1
We roughly calculated the execution time taken by each scheme for the same test. From the comparison results we found that the WENO5 and KL schemes take almost same amount of time to compute. In general, our comparison study indicates that the dimension-by-dimension WENO5 is very competitive as compared to the fully two-dimensional KL scheme in terms of accuracy and efficiency.
5. Summary and conclusions
Central-upwind finite-volume (CUFV) schemes are a class of Godunov-type method for solving hyperbolic conservation laws, and combine the nice features of the classical upwind and central FV methods. Semidiscrete central schemes are high-order accurate and nonoscillatory, depending on the reconstruction procedure, and these features make them computationally attractive for atmospheric numerical modeling. We consider semidiscretized high-order CUFV schemes with a dimension-by-dimension fifth-order WENO reconstruction (WENO5) and a fourth-order fully 2D (KL) reconstruction. The flux computations are based on flux formula introduced in Kurganov and Petrova (2001), which employs a compact approach and relies on local wind speed. Time integration is performed with a fourth-order Runge–Kutta method for the WENO5 and KL schemes.
The WENO-based schemes are only essentially nonoscillatory indicating that oscillations of small amplitude will still remain in the solution. In a strict sense WENO schemes are not positivity preserving. To address the positivity issue, a bound-preserving (BP) conservative filter is combined with WENO reconstructions, and a positivity-preserving (PP) filter is used. The BP and PP filters are local and computationally inexpensive. To compare these schemes we use several benchmark tests on the cubed-sphere geometry. The cubed-sphere geometry is a challenging computational domain for FV schemes, because of the nonorthogonal curvilinear geometry and grid discontinuities at the edges and corners. We used a 1D interpolation method to extend grid points (ghost cells) along the great-circle arc at the edges for computational stencils. This interpolation procedure combines quadratic and cubic-Lagrange interpolations and does not require a third panel at the corner ghost cell, which simplifies the implementation of the WENO5 and KL schemes.
The advection tests on the sphere include solid-body rotation of a cosine bell and moving (deforming) vortices. These two tests are quasi smooth; all the error norms show that the results with WENO5 and KL schemes are very close. In addition, a new challenging deformational flow test was also used to assess the performance of the nonoscillatory scheme in the presence of strong discontinuities. The BP and PP filter combination perform very well for the nonsmooth problem, and it does not degrade the accuracy when the problem is smooth. The execution time was roughly calculated using the WENO5 scheme as a basic reference, and it shows that KL scheme takes little less time to compute and produces similar results. The error norms suggest that the results with spherical WENO5 and KL are comparable to those published with recent high-order (global) FV schemes (Ullrich et al. 2010; Chen and Xiao 2008).
The 1D component of the WENO5 scheme is fifth-order accurate, nevertheless, the dimension-by-dimension approach may cause reduction in the formal order of accuracy of the resulting 2D scheme to second order. However, the empirical convergence rate for a smooth solid-body rotation test indicates that both the WENO5 and KL schemes maintain an order of accuracy between the third and fourth order. For a very challenging deformational flow test (Lauritzen et al. 2012) the order of accuracy further reduces, and is in between the second and third order. Unfortunately other high-order FV models (recently published) do not report empirical convergence results with the deformational flow tests.
In terms of practical implementation (algorithmic simplicity), WENO5 is a clear winner because the underlying computational stencil is simple and does not require corner ghost cells. The 1D method used for creating halo regions may not be the best choice, especially for high-order fully 2D FV schemes. However, a new method based on a Hermitian compact stencil is available (Croisille 2013) for cubed-sphere grids for high-order interpolations. We will further investigate this approach for our future applications. The Gaussian quadrature approach proposed by Ullrich et al. (2010) might be a good option for the 2D KL scheme, and is a topic for a future study. The benefits of BP and PP filters with CUFV schemes will be further studied for preservation of the tracer correlation and other desirable properties required for atmospheric chemistry applications (Lauritzen et al. 2012). It is not clear whether the WENO5 can perform better than a fully 2D scheme for nonlinear problems. This will be a matter for a future study, using a nonlinear global shallow-water model. Work in this direction is progressing and will be reported elsewhere.
Acknowledgments
The first author wishes to acknowledge Dr. Richard Loft for the SIParCS internship at IMAGe and Dr. Christopher Davis for the ASP graduate student visit opportunity at NCAR, both of which contributed to this research. Many thanks to Evan Bollig for helpful discussions. We thank Dr. Piotr Smolarkiewicz for giving in-depth details on the positivity-preserving filter. The authors gratefully acknowledge the internal review by Dr. Jeffrey S. Whitaker (NOAA/ESRL). Finally Kiran thanks Dr. Leticia Velazquez, Director of the CPS Program at UTEP, for the financial support provided during his doctoral studies. RDN is thankful to the U.S. DOE BER DE-SC0001658 for financial support.
APPENDIX A
2D KL Scheme Reconstruction Details



APPENDIX B
Constants for the SSP-RK(5,4) Scheme
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