An efficient quadrature rule on the Cubed Sphere

Portelenelle, Brice; Croisille, Jean‐Pierre

doi:10.1016/j.cam.2017.06.027

Cited by 7 publications

(12 citation statements)

References 13 publications

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“…Corollary 10. The quadrature rule Q N exactly integrates 15 16 of all real Legendre spherical harmonics. More precisely, for all |m| ≤ n,…”

Section: ∀Q ∈ G ∀U ∈ Umentioning

confidence: 99%

“…The ratio 15/16 of the real Legendre basis is asymptotic. In [7,15], a similar approach based on invariance properties provided an asymptotic ratio of 7/8 of the complex Legendre basis. We observe here that using the real Legendre basis highlights further exact results.…”

Section: ∀Q ∈ G ∀U ∈ Umentioning

confidence: 99%

“…Here the Cubed Sphere nodes are selected from the beginning as good quadrature nodes, and therefore, only the weights have to be identied. In [15], two examples of weights have been suggested. The rst one was based on some extended trapezoidal rule, attributing some area to each node.…”

Section: Introductionmentioning

confidence: 99%

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Quadrature and symmetry on the Cubed Sphere

Bellet

Brachet

Croisille

2022

Journal of Computational and Applied Mathematics

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In the companion paper [6], a spherical harmonics subspace associated to the Cubed Sphere has been introduced. This subspace is further analyzed here. In particular, it permits to dene a new Cubed Sphere based quadrature. This quadrature inherits the rotationally invariant properties of the spherical harmonics subspace. Contrary to Gauss quadratures, where the set of nodes and weights is solution of a nonlinear system, only the weights are unknown here. Despite this conceptual simplicity, the new quadrature displays an accuracy comparable to optimal quadratures, such as the Lebedev rules.

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“…Corollary 10. The quadrature rule Q N exactly integrates 15 16 of all real Legendre spherical harmonics. More precisely, for all |m| ≤ n,…”

Section: ∀Q ∈ G ∀U ∈ Umentioning

confidence: 99%

Section: ∀Q ∈ G ∀U ∈ Umentioning

confidence: 99%

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Quadrature and symmetry on the Cubed Sphere

Bellet

Brachet

Croisille

2022

Journal of Computational and Applied Mathematics

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“…All the integrals in () are evaluated with the integral

\int_{𝕊_{a}^{2}} f (x) d σ (x)

approximated by Q N ( f ), the rule (20) in Portelenelle and Croisille (2018). This rule is defined by

Q_{N} (f) = a^{2} \sum_{k = false(I false)}^{false(V I false)} \sum_{i, j = prefix- N false/ 2}^{N false/ 2} α_{i, j} F false(s_{i, j}^{k} false) .

Portelenelle and Croisille (2018) define the weights

α_{i, j}

.…”

Section: Conservationmentioning

confidence: 99%

“…All the integrals in (31) are evaluated with the integral ∫ S 2 a f (x) d (x) approximated by Q N (f ), the rule (20) in Portelenelle and Croisille (2018). This rule is defined by…”

Section: Conservation With the Hccs Schemementioning

confidence: 99%

Spherical shallow‐water wave simulation by a cubed‐sphere finite‐difference solver

Brachet

Croisille

2020

Quart J Royal Meteoro Soc

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We consider the test suite for the shallow‐water (SW) equations on the sphere suggested by Paldor in earlier work. This series of tests consists of zonally propagating wave solutions on the full sphere. Two series of solutions are considered. The first series is referred to as “barotropic". It consists of an extension of the Rossby–Haurwitz test case. The second series, referred to as “baroclinic”, consists of a generalisation of the Matsuno solution to the linearized SW equations in an equatorial channel. The Hermitian Compact Cubed Sphere (HCCS) model which is used in this paper is a recently introduced SW solver on the sphere. The spatial approximation is a centred finite‐difference scheme based on high‐order differencing along great circles. The time stepping is performed by the explicit RK4 scheme or by an exponential scheme. For both barotropic and baroclinic test case series, the results show a very good agreement of the numerical solution with the analytic one, even for long time simulations.

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ARPIST: Provably accurate and stable numerical integration over spherical triangles

Jiao

2023

Journal of Computational and Applied Mathematics

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An efficient quadrature rule on the Cubed Sphere

Cited by 7 publications

References 13 publications

Quadrature and symmetry on the Cubed Sphere

Quadrature and symmetry on the Cubed Sphere

Spherical shallow‐water wave simulation by a cubed‐sphere finite‐difference solver

ARPIST: Provably accurate and stable numerical integration over spherical triangles

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