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Ramachandran D. Nair

Abstract

A second-order diffusion scheme is developed for the discontinuous Galerkin (DG) global shallow-water model. The shallow-water equations are discretized on the cubed sphere tiled with quadrilateral elements relying on a nonorthogonal curvilinear coordinate system. In the viscous shallow-water model the diffusion terms (viscous fluxes) are approximated with two different approaches: 1) the element-wise localized discretization without considering the interelement contributions and 2) the discretization based on the local discontinuous Galerkin (LDG) method. In the LDG formulation the advection–diffusion equation is solved as a first-order system. All of the curvature terms resulting from the cubed-sphere geometry are incorporated into the first-order system. The effectiveness of each diffusion scheme is studied using the standard shallow-water test cases. The approach of element-wise localized discretization of the diffusion term is easy to implement but found to be less effective, and with relatively high diffusion coefficients, it can adversely affect the solution. The shallow-water tests show that the LDG scheme converges monotonically and that the rate of convergence is dependent on the coefficient of diffusion. Also the LDG scheme successfully eliminates small-scale noise, and the simulated results are smooth and comparable to the reference solution.

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Ramachandran D. Nair

Abstract

The conservative cascade scheme (CCS) combines a one-dimensional mass-conserving finite-volume method with an efficient semi-Lagrangian scheme over the sphere. Major limitations of this scheme are a breakdown in the polar region due to the coordinate singularity and a restriction to polar meridional Courant number C θ ≤ 1. The proposed scheme is an extension of the CCS over the sphere for C θ > 1. This is achieved by isolating a region near the pole where the CCS fails and applying a two-dimensional remapping based on the cell-integrated semi-Lagrangian (CISL) scheme. The interface between these two schemes is the singular polar region where the total mass is computed and redistributed in a conservative manner. The resulting scheme is applicable to large polar Courant number and is tested using solid-body rotation and a deformational flow.

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Ramachandran D. Nair

Abstract

The discontinuous Galerkin (DG) discretization relies on an integral (weak) formulation of the hyperbolic conservation law, which leads to the evaluation of several surface and line integrals for multidimensional problems. An alternative formulation of the DG method is possible under the flux reconstruction (FR) framework, where the equations are solved in differential form and the discretization is free from quadrature rules, resulting in computationally efficient algorithms. The author has implemented a quadrature-free form of the nodal DG method based on the FR approach combined with spectral differencing (SD), in a shallow-water (SW) model employing cubed-sphere geometry. The performance of the SD model is compared with the regular nodal DG variant of the SW model using several benchmark tests, including a viscous test case. A positivity-preserving local filter is tested for SD advection that removes spurious oscillations while being conservative and accurate. In this implementation, the SD formulation is found to be 18% faster than the DG method for inviscid SW tests cases and 24% faster for the viscous case. The results obtained by the SD formulation are on par with the regular nodal DG formulation in terms of accuracy and convergence.

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Michael D. Toy
and
Ramachandran D. Nair

Abstract

An energy and potential enstrophy conserving finite-difference scheme for the shallow-water equations is derived in generalized curvilinear coordinates. This is an extension of a scheme formulated by Arakawa and Lamb for orthogonal coordinate systems. The starting point for the present scheme is the shallow-water equations cast in generalized curvilinear coordinates, and tensor analysis is used to derive the invariant conservation properties. Preliminary tests on a flat plane with doubly periodic boundary conditions are presented. The scheme is shown to possess similar order-of-convergence error characteristics using a nonorthogonal coordinate compared to Cartesian coordinates for a nonlinear test of flow over an isolated mountain. A linear normal mode analysis shows that the discrete form of the Coriolis term provides stationary geostrophically balanced modes for the nonorthogonal coordinate and no unphysical computational modes are introduced. The scheme uses centered differences and averages, which are formally second-order accurate. An empirical test with a steady geostrophically balanced flow shows that the convergence rate of the truncation errors of the discrete operators is second order. The next step will be to adapt the scheme for use on the cubed sphere, which will involve modification at the lateral boundaries of the cube faces.

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Ramachandran D. Nair
and
Christiane Jablonowski

Abstract

A new two-dimensional advection test on the surface of the sphere is proposed. The test combines a solid-body rotation and a deformational flow field to form moving vortices over the surface of the sphere. The resulting time-dependent deforming vortex centers are located on diametrically opposite sides of the sphere and move along a predetermined great circle trajectory. The horizontal wind field is deformational and nondivergent, and the analytic solution is known at any time. During one revolution around the sphere the initially smooth transported scalar develops strong gradients. Such an approach is therefore more challenging than existing advection test cases on the sphere. To demonstrate the effectiveness and versatility of the proposed test, three different advection schemes are employed, such as a discontinuous Galerkin method on a cubed-sphere mesh, a classical semi-Lagrangian method, and a finite-volume algorithm with adaptive mesh refinement (AMR) on a regular latitude–longitude grid. The numerical results are compared with the analytic solution for different flow orientation angles on the sphere.

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Ramachandran D. Nair
and
Bennert Machenhauer

Abstract

A mass-conservative cell-integrated semi-Lagrangian (CISL) scheme is presented and tested for 2D transport on the sphere. The total mass is conserved exactly and the mass of each individual grid cell is conserved in general. The scheme is based on a general scheme developed by Machenhauer and Olk that has increased cost effectiveness without loss of accuracy, compared to the CISL scheme of Rančić. A regular latitude–longitude grid is used on the sphere and upstream trajectories from the corner points of the regular grid cells (the Eulerian cells) define the corner points of the departure cells. The sides in these so-called Lagrangian cells are generally defined as straight lines in a (λ, μ) plane, where λ is the longitude and μ is the sine of the latitude. The mass distribution within each Eulerian grid cell is defined by quasi-biparabolic functions, which are used to integrate analytically the mass in each Lagrangian computational cell. The auxiliary computational cells are polygons with each side parallel to the coordinate axis. Also, the computational cells have the same area as the Lagrangian cells they approximate. They were introduced in order to simplify the analytical integrals of mass. Near the poles, the east and west sides of certain Lagrangian cells cannot be approximated by straight lines in the (λ, μ) plane, and are instead represented by straight lines in polar tangent plane coordinates. Each of the latitudinal belts of Lagrangian cells in the polar caps are split up into several latitudinal belts of subcells, which can be approximated by computational cells as in the case of cells closer to the equator. One latitudinal belt in each hemisphere, which encloses the Eulerian pole (singular belt), is treated in a special way. First the total mass in the singular belt is determined and then it is redistributed among the cells in the belt using weights determined by a traditional SL scheme at the midpoints of the cells. By this procedure the total mass is still conserved while the conservation is only approximately maintained for the individual cells in the singular belt. These special treatments in the polar caps fit well into the general structure of the code and can be implemented with minor modifications in the code used for the rest of the sphere.

Compared to two other conservative advection schemes implemented on the sphere the CISL scheme used here was found to be competitive in terms of accuracy for the same resolution. In addition the CISL scheme has the advantage over these schemes that it is applicable for Courant numbers larger than one. In plane geometry the scheme of Rančić had an overhead factor of 2.5 in CPU time compared to a traditional bicubic semi-Lagrangian scheme. This factor is reduced to 1.1 for the Machenhauer and Olk scheme on the plane while on the sphere the factor is found to be 1.28 for the present scheme. This overhead seems to be a reasonable price to pay for increased accuracy and exact mass conservation.

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Peter H. Lauritzen
and
Ramachandran D. Nair

Abstract

A high-order monotone and conservative cascade remapping algorithm between spherical grids (CaRS) is developed. This algorithm is specifically designed to remap between the cubed-sphere and regular latitude–longitude grids. The remapping approach is based on the conservative cascade method in which a two-dimensional remapping problem is split into two one-dimensional problems. This allows for easy implementation of high-order subgrid-cell reconstructions as well as the application of advanced monotone filters. The accuracy of CaRS is assessed by remapping analytic fields from the regular latitude–longitude grid to the cubed-sphere grid. In terms of standard error measures, CaRS is found to be competitive relative to an existing algorithm when regridding from a fine to a coarse grid and more accurate when regridding from a coarse to a fine grid.

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Matthew R. Norman
and
Ramachandran D. Nair

Abstract

A group of new conservative remapping schemes based on nonpolynomial approximations is proposed. The remapping schemes rely on the conservative cascade scheme (CCS), which employs an efficient sequence of 1D remapping operations to solve a multidimensional problem. The present study adapts three new nonpolynomial-based reconstructions of subgrid variation to the CCS: the Piecewise Hyperbolic Method (PHM), the Piecewise Double Hyperbolic Method (PDHM), and the Piecewise Rational Method (PRM) for comparison with the baseline method: the Piecewise Parabolic Method (PPM). Additionally, an adaptive hybrid approximation scheme, PPM-Hybrid (PPM-H), is constructed using monotonic PPM for smooth data and local extrema and using PHM for steep jumps where PPM typically suffers large accuracy degradation because of its original monotonic filter. Smooth and nonsmooth data profiles are transported in 1D, 2D Cartesian, and 2D spherical frameworks under uniform advection, solid-body rotation, and deformational flow. Accuracy is compared via the L 1 global error norm. In general, PPM outperformed PHM, but when the majority of the error came from PPM degradation at sharp derivative changes (e.g., the vicinity near sine wave extrema), PHM was more accurate. PRM performed very similarly to PPM for nonsmooth functions, but the order of convergence was worse than PPM for smoother data. PDHM performed the worst of all of the nonpolynomial methods for nearly every test case. PPM-H outperformed PPM and all of the nonpolynomial methods for all test cases in all geometries, offering a robust advantage in the CCS scheme with a negligible increase in computational time.

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David M. Hall
and
Ramachandran D. Nair

Abstract

A discontinuous Galerkin (DG) transport scheme is presented that employs the Yin–Yang grid on the sphere. The Yin–Yang grid is a quasi-uniform overset mesh comprising two notched latitude–longitude meshes placed at right angles to each other. Surface fluxes of conserved scalars are obtained at the overset boundaries by interpolation from the interior of the elements on the complimentary grid, using high-order polynomial interpolation intrinsic to the DG technique. A series of standard tests are applied to evaluate its performance, revealing it to be robust and its accuracy to be competitive with other global advection schemes at equivalent resolutions. Under p-type grid refinement, the DG Yin–Yang method exhibits spectral error convergence for smooth initial conditions and third-order geometric convergence for C 1 continuous functions. In comparison with finite-volume implementations of the Yin–Yang mesh, the DG implementation is less complex, as it does not require a wide halo region of elements for accurate boundary value interpolation. With respect to DG cubed-sphere implementations, the Yin–Yang grid exhibits similar accuracy and appears to be a viable alternative suitable for global advective transport. A variant called the Yin–Yang polar (YY-P) mesh is also examined and is shown to have properties similar to the original Yin–Yang mesh while performing better on tests with strictly zonal flow.

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Yifan Zhang
and
Ramachandran D. Nair

Abstract

The discontinuous Galerkin (DG) method is high order, conservative, and offers excellent parallel efficiency. However, when there are discontinuities in the solution, the DG transport scheme generates spurious oscillations that reduce the quality of the numerical solution. For applications such as the atmospheric tracer transport modeling, a nonoscillatory, positivity-preserving solution is a basic requirement. To suppress the oscillations in the DG solution, a limiter based on the Hermite-Weighted Essentially Nonoscillatory (H-WENO) method has been implemented for a third-order DG transport scheme. However, the H-WENO limiter can still produce wiggles with small amplitudes in the solutions, but this issue has been addressed by combining the limiter with a bound-preserving (BP) filter. The BP filter is local and easy to implement and can be used for making the solution strictly positivity preserving. The DG scheme combined with the limiter and filter preserves the accuracy of the numerical solution in the smooth regions while effectively eliminating overshoots and undershoots. The resulting nonoscillatory DG scheme is third-order accurate (P 2-DG) and based on the modal discretization. The 2D Cartesian scheme is further extended to the cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates. The accuracy and effectiveness of the limiter and filter are demonstrated with several benchmark tests on both the Cartesian and spherical geometries.

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