The “Cubed Sphere”: A New Method for the Solution of Partial Differential Equations in Spherical Geometry

Ronchi, C.; Iacono, Roberto; Paolucci, P.

doi:10.1006/jcph.1996.0047

Cited by 444 publications

(431 citation statements)

References 9 publications

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“…1). An angularly equidistant mapping [18] is used to generate the initial grids of the six adjoining grid sectors (or panels) that seamlessly cover each of the 2D spheres. In contrast to flows on 2D cubed-sphere grids, for which a curved coordinate system is normally defined on each of the six cubed-sphere sectors, the 3D cubed-sphere grid in principle allows the use of a unique coordinate system (e.g., Cartesian) to discretize the governing conservation laws everywhere in the physical domain, which makes unnecessary the usage of a covariant transformation [18,22,41,42] to map vector fields from the curved coordinate system to the Cartesian system.…”

Section: Glm Formulation Of Ideal Mhd Governing Equationsmentioning

confidence: 99%

“…In recent years, cubed-sphere grids have gained increasing popularity for simulating fluid flow in domains between concentric spheres, first in the area of climate and weather modelling [18,19,20,21,22,23,24], but more recently also in areas like astrophysics [25,26]. Very recently, Ivan et al [14,15] have proposed a second-order parallel solution-adaptive computational framework for solving hyperbolic conservation laws on 3D cubed-sphere grids and applied the formulation to the simulation of several magnetized and nonmagnetized space-physics problems.…”

Section: Introductionmentioning

confidence: 99%

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High-order central ENO finite-volume scheme for ideal MHD

Susanto

Ivan

Sterck

et al. 2013

Journal of Computational Physics

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A high-order central essentially non-oscillatory (CENO) finite-volume scheme is developed for the compressible ideal magnetohydrodynamics (MHD) equations solved on threedimensional (3D) cubed-sphere grids. The proposed formulation is an extension to 3D geometries of a recent high-order MHD CENO scheme developed on two-dimensional (2D) grids. The main technical challenge in extending the 2D method to 3D cubed-sphere grids is to properly handle the nonplanar cell faces that arise in cubed-sphere grids. This difficulty is solved by considering general hexahedral cells with trilinear faces, which allow us to compute fluxes, areas and volumes with high-order accuracy by transforming to a reference cubic cell. The 3D CENO scheme is implemented within a flexible multi-block cubed-sphere grid framework to fourth-order accuracy, resulting in a high-order solution method for cubed-sphere grids with unique capabilities in terms of adaptive refinement and parallel scalability. The high-order method is applicable to the solution of general hyperbolic conservation laws with, in principle, arbitrary order. The CENO scheme is based on a hybrid solution reconstruction procedure that provides high-order accuracy in smooth regions, even for smooth extrema, and non-oscillatory transitions at discontinuities. The scheme is applied herein to MHD in combination with a GLM divergence correction technique to control the solenoidal condition for the magnetic field while preserving the high-order accuracy of the numerical procedure. The cubed-sphere simulation framework features a flexible design based on a genuine multi-block implementation, leading to high-order accuracy, flux calculation, adaptivity and parallelism that are fully transparent to the boundaries between the six sectors of the cubedsphere grid. The proposed 3D MHD CENO scheme is shown to achieve uniform fourth-order accuracy on cubed-sphere grids. Parallel domain partitioning and grid adaptivity are achieved on the 3D cubed-sphere grids using a hierarchical block-based division strategy with blocks of equal size. Numerical results to demonstrate the high-order accuracy, robustness and capability of the proposed high-order framework are presented and discussed for several test problems.

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Section: Glm Formulation Of Ideal Mhd Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-order central ENO finite-volume scheme for ideal MHD

Susanto

Ivan

Sterck

et al. 2013

Journal of Computational Physics

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“…Explicit expressions for the curl, gradient, and divergence using the physical components of a vector are given in Ronchi et al [1].…”

Section: Basic Equationsmentioning

confidence: 99%

“…This code was capable of handling high viscosity contrasts in a spherical shell, however its usefulness was limited by the coordinate singularity at the poles. Ronchi et al [1] proposed a new method for the solution of partial differential equations in spherical geometry consisting of projecting a cube onto a sphere to produce six separate patches that are then subdivided. The coordinate lines on each patch correspond to great circles that are perpendicular at the centers of each block (see figure 1).…”

Section: Introductionmentioning

confidence: 99%

“…

AbstractWe present a new finite difference code for modeling three-dimensional thermal convection in a spherical shell using the "cubed sphere" method of Ronchi et al [1]. The equation of motion is solved using a poloidal potential formulation of the equation of motion for an iso-viscous, infinite Prandtl number fluid on a finite difference grid and advective transport is implemented using the 2 nd -order MPDATA scheme of Smolarkiewicz [2].

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Three-dimensional spherical shell convection at infinite Prandtl number using the ‘cubed sphere’ method

HERNLUND¹,

TACKLEY

2003

Computational Fluid and Solid Mechanics 2003

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We present a new finite difference code for modeling three-dimensional thermal convection in a spherical shell using the "cubed sphere" method of Ronchi et al. [1]. The equation of motion is solved using a poloidal potential formulation of the equation of motion for an iso-viscous, infinite Prandtl number fluid on a finite difference grid and advective transport is implemented using the 2 nd -order MPDATA scheme of Smolarkiewicz [2]. Due to the high efficiency of multigrid acceleration, low memory requirements, and second-order accuracy of this model, we conclude that the cubed sphere method offers a great deal of potential for simulating complicated problems of fluid mechanics in spherical geometry. IntroductionHighly viscous thermal convection in the rocky mantles of the solid planets is the primary process governing their thermal and mechanical evolution over long time scales. This process is driven primarily by the transfer of heat from the interior to the surface. Because thermal convection constitutes a non-linear problem, the main tool for studying finite amplitude convective motions is computer models. Due to the relative simplicity of the relevant equations in a parallel coordinate system, Cartesian domains are often used to obtain an approximation to convective motions in planetary mantles. This approximation has proven to be useful for many different problems, although it imposes a geometric symmetry between the upper and lower boundaries which does not exist in spherical geometry.In the first spherical shell models the most popular method for solving the relevant equations were spectral, i.e., using spherical harmonics as basis functions. This allows a great deal of simplification of the equations (Chandrasekhar [3]), however, computation of the Legendre transforms can be expensive. The method's primary limitation, however, is that the elegance and simplicity are destroyed when more complicated effects, such as laterally-varying viscosity, are included, and viscosity varies by many orders of magnitude in planetary bodies. In addition, spectral methods do not offer a straightforward implementation on parallel computers since the basis functions are not local.Recently, there has been an increase in the application of grid-based methods to mantle convection in spherical geometry. Methods such as finite elements and finite differences are local and can therefore more easily accommodate complex effects such as

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Non-reflecting boundary conditions for electromagnetic scattering

Grote

2000

Int. J. Numer. Model.

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An exact nonre ecting boundary condition was derived previously for use with the time dependent Maxwell equations in three space dimensions 1. Here it is shown how to combine that boundary condition with the variational formulation for use with the nite element method. The fundamental theory underlying the derivation of the exact boundarycondition is reviewed. Numerical examples with the nite-di erence timedomain method are presented which demonstrate the high accuracy and con rm the expected rate of convergence of the numerical method.

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The “Cubed Sphere”: A New Method for the Solution of Partial Differential Equations in Spherical Geometry

Cited by 444 publications

References 9 publications

High-order central ENO finite-volume scheme for ideal MHD

High-order central ENO finite-volume scheme for ideal MHD

Three-dimensional spherical shell convection at infinite Prandtl number using the ‘cubed sphere’ method

Non-reflecting boundary conditions for electromagnetic scattering

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