Interpolating data on the Cubed Sphere with Spherical Harmonics

2021 ◽  
Author(s):  
Jean-Pierre Croisille ◽  
Jean-Baptiste Bellet ◽  
Matthieu Brachet
Keyword(s):  
Numerical Simulation ◽  
Symmetry Group ◽  
Spherical Harmonics ◽  
Recent Progress ◽  
Quadrature Rules ◽  
Geometrical Properties ◽  
Truncation Scheme ◽  
Cubed Sphere

<p>The Cubed Sphere is a grid commonly used in numerical simulation in climatology. In this talk we present recent progress<br>on the algebraic and geometrical properties of this highly symmetrical grid.<br>First, an analysis of the symmetry group of the Cubed Sphere will be presented: this group <br>is identified as the group of the Cube, [1]. Furthermore, we show how to construct a discrete Spherical Harmonics (SH) basis associated to <br>the Cubed Sphere. This basis displays a truncation scheme relating the zonal and longitudinal <br>mode numbers reminiscent of the rhomboidal truncation on the Lon-Lat grid.<br>The new analysis allows to derive new quadrature rules of  interest for applications in any kind of spherical modelling. In addition,<br>we will comment on applications in mathematical climatology and meteorology, [2].</p><p>[1] J.-B. Bellet, Symmetry group of the equiangular Cubed Sphere, preprint, IECL, Univ. Lorraine, 2020, submitted</p><p>[2] J.-B. Bellet, M. Brachet and J.-P. Croisille, Spherical Harmonics on The Cubed Sphere, IECL, Univ. Lorraine, 2021, Preprint.</p>

scholarly journals Formulation of Macroscopic Transport Models for Numerical Simulation of Semiconductor Devices

VLSI Design ◽  
1995 ◽  
Vol 3 (2) ◽  
pp. 211-224 ◽  
Author(s):  
Edwin C. Kan ◽  
Zhiping Yu ◽  
Robert W. Dutton ◽  
Datong Chen ◽  
Umberto Ravaioli
Keyword(s):  
Numerical Simulation ◽  
Computational Efficiency ◽  
Recent Progress ◽  
Energy Transport ◽  
Semiconductor Devices ◽  
Boltzmann Transport Equation ◽  
Boltzmann Transport ◽  
Drift Diffusion ◽  
Thermoelectric Effects ◽  
Transport Models

According to different assumptions in deriving carrier and energy flux equations, macroscopic semiconductor transport models from the moments of the Boltzmann transport equation (BTE) can be divided into two main categories: the hydrodynamic (HD) model which basically follows Bløtekjer's approach [1, 2], and the Energy Transport (ET) model which originates from Strattton's approximation [3, 4]. The formulation, discretization, parametrization and numerical properties of the HD and ET models are carefully examined and compared. The well-known spurious velocity spike of the HD model in simple nin structures can then be understood from its formulation and parametrization of the thermoelectric current components. Recent progress in treating negative differential resistances with the ET model and extending the model to thermoelectric simulation is summarized. Finally, we propose a new model denoted by DUET (Dual ET)which accounts for all thermoelectric effects in most modern devices and demonstrates very good numerical properties. The new advances in applicability and computational efficiency of the ET model, as well as its easy implementation by modifying the conventional drift-diffusion (DD) model, indicate its attractiveness for numerical simulation of advanced semiconductor devices

scholarly journals Comparisons of the Cubed-Sphere Gravity Model with the Spherical Harmonics

2010 ◽  
Vol 33 (2) ◽  
pp. 415-425 ◽  
Author(s):  
Brandon A. Jones ◽  
George H. Born ◽  
Gregory Beylkin
Keyword(s):  
Gravity Model ◽  
Spherical Harmonics ◽  
Cubed Sphere

On the symmetries of spherical harmonics

1957 ◽  
Vol 53 (2) ◽  
pp. 343-367 ◽  
Author(s):  
S. L. Altmann
Keyword(s):  
Irreducible Representation ◽  
Band Structure ◽  
Symmetry Group ◽  
Spherical Harmonics ◽  
Lower Order ◽  
Central Atom ◽  
Real Form ◽  
Cellular Method ◽  
Hybrid Orbitals

It is often necessary to obtain expansions in spherical harmonics that belong to a given irreducible representation of a symmetry group. This is the case, for instance, when the cellular method is applied to investigate the band structure of a metal, where expansions are required that reproduce the symmetry of the group of the k vector (see Bouckaert, Smoluchowski and Wigner(4); von der Lage and Bethe (9)). Another instance where such expansions are necessary appears when hybrid orbitals are obtained for a central atom in a molecule of given symmetry (Kimball (8)). In this case lower order spherical harmonics are considered and tables for them up to l = 2 (functions s, p and d in real form) are given in the literature (cf. for example Eyring, Walter and Kimball (5)). However, interest has recently arisen in hybrids that include f functions (Shirmazan and Dyatkina (12)) and an extension of these tables appears to be desirable.

scholarly journals Weakly Orthogonal Spherical Harmonics in a Non-Polar Spherical Coordinates and its Application to Functions on Cubed-Sphere

2012 ◽  
Vol 17 (2) ◽  
pp. 200
Author(s):  
M.A.A.M. Faham ◽  
H.M. Nasir
Keyword(s):  
Fourier Series ◽  
Coordinate System ◽  
Spherical Harmonics ◽  
Spherical Coordinate System ◽  
Spherical Functions ◽  
Spherical Coordinates ◽  
Spherical Coordinate ◽  
Linear Relations ◽  
Inner Products ◽  
Cubed Sphere

In a recent paper (Nasir, 2007), a set of weakly orthogonal and completely orthogonal spherical harmonics in a non-polar spherical coordinate system based on a cubed-sphere was constructed. In this work, we explore some linear relations between these two sets of spherical harmonics. Moreover, a power representation for the set of weakly orthogonal spherical harmonics corresponding to a mode is presented. We also determine the norm of the orthogonal spherical harmonics and hence the inner products for the weakly orthogonal spherical harmonics. As an immediate application of these properties, we present a Fourier series formulation of spherical functions defined on the cubed-sphere.  

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