Non-orthogonal version of the arbitrary polygonal C-grid and a new diamond grid

Quasi-uniform grids of the sphere have become popular recently since they avoid parallel scaling bottle- necks associated with the poles of latitude–longitude grids. However quasi-uniform grids of the sphere are often non- orthogonal. A version of the C-grid for arbitrary non- orthogonal grids is presented which gives some of the mimetic properties of the orthogonal C-grid. Exact energy conservation is sacrificed for improved accuracy and the re- sulting scheme numerically conserves energy and potential enstrophy well. The non-orthogonal nature means that the scheme can be used on a cubed sphere. The advantage of the cubed sphere is that it does not admit the computa- tional modes of the hexagonal or triangular C-grids. On var- ious shallow-water test cases, the non-orthogonal scheme on a cubed sphere has accuracy less than or equal to the orthog- onal scheme on an orthogonal hexagonal icosahedron. A new diamond grid is presented consisting of quasi- uniform quadrilaterals which is more nearly orthogonal than the equal-angle cubed sphere but with otherwise similar properties. It performs better than the cubed sphere in ev- ery way and should be used instead in codes which allow a flexible grid structure

An efficient quadrature rule on the cubed sphere

A new quadrature rule for functions defined on the sphere is introduced. The nodes are defined as the points of the Cubed Sphere. The associated weights are defined in analogy to the trapezoidal rule on each panel of the Cubed Sphere. The formula enjoys a symmetry property ensuring that a proportion of 7/8 of all Spherical Harmonics is integrated exactly. Based on the remaining Spherical Harmonics, it is possible to define modified weights giving an enhanced quad-rature rule. Numerical results show that the new quadrature is competitive with classical rules of the litterature. This second quadrature rule is believed to be of interest for applied mathematicians, physicists and engineers dealing with data located at the points of the Cubed Sphere

A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed Sphere

In a previous article [J. Comp. Phys. $\mathbf{357}$ (2018) 282-304], the mixed mimetic spectral element method was used to solve the rotating shallow water equations in an idealized geometry. Here the method is extended to a smoothly varying, non-affine, cubed sphere geometry. The differential operators are encoded topologically via incidence matrices due to the use of spectral element edge functions to construct tensor product solution spaces in $H(\mathrm{rot})$, $H(\mathrm{div})$ and $L_2$. These incidence matrices commute with respect to the metric terms in order to ensure that the mimetic properties are preserved independent of the geometry. This ensures conservation of mass, vorticity and energy for the rotating shallow water equations using inexact quadrature on the cubed sphere. The spectral convergence of errors are similarly preserved on the cubed sphere, with the generalized Piola transformation used to construct the metric terms for the physical field quantities

Fourier Series in a Non-polar Coordinate System

Numerical computations of functions in a grid on the surface of a sphere and the integration of corresponding partial differential equation are very important tasks among the researchers in the field of computational mathematics. Although the spherical polar coordinate system is the most elegant tool in this respect, the problems due to polar concentration near the North and South poles brings much difficulties to the computational scientists. Cubed sphere concept is one of the techniques used recently in the processing of searching remedies for such problems. In earlier works, the second author of this article has constructed weakly and fully orthogonal spherical harmonics in a non-polar coordinate system developed based on the 'cubed-sphere' concept Then, we established relational properties between the two sets of spherical harmonics and between the functions defined in the six faces of the cubed sphere. Fourier series techniques could be applied to a wide array of mathematical and physical problems. In mis work, we express the Fourier series of a spherical function in terms of the weakly orthogonal spherical harmonics of the non-polar coordinate system. Also, we discuss the method of finding the Fourier series coefficients efficiently using the relational properties established

An application of Fourier series expansion of a function in a non-polar spherical coordinate system

Cubed sphere is one of the main tools used to avoid pole problems those arise in the selection of spherical polar coordinates. In this respect, earlier we considered a recently developed cubed sphere based on coordinate mapping over the cubed surface. The function on the sphere was treated as an ordered set of six-tuples. In that work, we established weakly orthogonal and completely orthogonal spherical harmonics of the system and developed corresponding symmetric and linear relations. Also, we found the norm of the orthogonal spherical harmonics. In this work, we explore the Fourier representation of a spherical function on this coordinate system in terms of weakly orthogonal spherical harmonics. The advantages of the linear relation between the two sets of spherical harmonics and diagonal property of the norm of the fully orthogonal spherical harmonics were in cooperated for this work. We also strength our work by giving an example to demonstrate how Fourier coefficients can be computed to represent a given spherical function in terms of the spherical harmonics of the coordinate system

The Representation of Tropical Cyclones Within the Global William Putman Non-Hydrostatic Goddard Earth Observing System Model (GEOS-5) at Cloud-Permitting Resolutions

The Goddard Earth Observing System Model (GEOS-S), an earth system model developed in the NASA Global Modeling and Assimilation Office (GMAO), has integrated the non-hydrostatic finite-volume dynamical core on the cubed-sphere grid. The extension to a non-hydrostatic dynamical framework and the quasi-uniform cubed-sphere geometry permits the efficient exploration of global weather and climate modeling at cloud permitting resolutions of 10- to 4-km on today's high performance computing platforms. We have explored a series of incremental increases in global resolution with GEOS-S from irs standard 72-level 27-km resolution (approx.5.5 million cells covering the globe from the surface to 0.1 hPa) down to 3.5-km (approx. 3.6 billion cells)

Application of the Cubed-Sphere Grid to Tilted Black-Hole Accretion Disks

In recent work we presented the first results of global general relativistic magnetohydrodynamic (GRMHD) simulations of tilted (or misaligned) accretion disks around rotating black holes. The simulated tilted disks showed dramatic differences from comparable untilted disks, such as asymmetrical accretion onto the hole through opposing "plunging streams" and global precession of the disk powered by a torque provided by the black hole. However, those simulations used a traditional spherical-polar grid that was purposefully underresolved along the pole, which prevented us from assessing the behavior of any jets that may have been associated with the tilted disks. To address this shortcoming we have added a block-structured "cubed-sphere" grid option to the Cosmos++ GRMHD code, which will allow us to simultaneously resolve the disk and polar regions. Here we present our implementation of this grid and the results of a small suite of validation tests intended to demonstrate that the new grid performs as expected. The most important test in this work is a comparison of identical tilted disks, one evolved using our spherical-polar grid and the other with the cubed-sphere grid. We also demonstrate an interesting dependence of the early-time evolution of our disks on their orientation with respect to the grid alignment. This dependence arises from the differing treatment of current sheets within the disks, especially whether they are aligned with symmetry planes of the grid or not.Comment: 15 pages, 11 figures, submitted to Ap

Strategies for the characteristic extraction of gravitational waveforms

We develop, test, and compare new numerical and geometrical methods for improving the accuracy of extracting waveforms using characteristic evolution. The new numerical method involves use of circular boundaries to the stereographic grid patches which cover the spherical cross sections of the outgoing null cones. We show how an angular version of numerical dissipation can be introduced into the characteristic code to damp the high frequency error arising form the irregular way the circular patch boundary cuts through the grid. The new geometric method involves use of the Weyl tensor component Psi4 to extract the waveform as opposed to the original approach via the Bondi news function. We develop the necessary analytic and computational formula to compute the O(1/r) radiative part of Psi4 in terms of a conformally compactified treatment of null infinity. These methods are compared and calibrated in test problems based upon linearized waves

Relational properties of weakly orthogonal and orthogonal spherical harmonics in cubed sphere

Numerical computations on the sphere in solving problems defined on the sphere suffer from many difficulties near the poles when using spherical polar coordinate system for the spherical surface. For example, in the computations of global weather prediction models, concentrated grid points near the poles increase the amount of computations in the pole region where quantities of interest are of less important than in other parts of the globe. Such problems are collectively called as the 'pole problems'. Avoiding pole problems have attracted some researches in the recent past. One of the recent development in this direction is to define grid meshes on the sphere which do not contain polar concentrated points. Among these the 'cubed sphere' defined from the surface of a unit cube has been used by some authors for approximating weather prediction models by finite difference and finite element methods. In a recent paper, one of the present authors has constructed weakly orthogonal spherical harmonics in a non-polar spherical co-ordinate system based on the cube sphere concept. This can be used for approximating functions on the sphere by spectral methods without the pole problems. In this work, we establish some Lin ear and recurrence relations between these two sets of spherical harmonics. We also exploit linear relations between harmonic components defined in the six faces of the cubed sphere