Hermitian approximation of the spherical divergence on the Cubed-Sphere

Croisille, Jean‐Pierre

doi:10.1016/j.cam.2014.11.047

Cited by 4 publications

(6 citation statements)

References 25 publications

(55 reference statements)

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“…In this section, the Hermitian Compact Scheme on the Cubed Sphere (HCCS) is summarized (Croisille 2013, 2015; Brachet 2018; Brachet and Croisille 2019). Consider the spherical SW equations:

(SW) \{\begin{array}{l} \frac{\partial h^{⋆}}{\partial t} false(t, boldx false) + {normal∇}_{normalT} \cdot {h^{⋆} false(t, boldx false) boldu false(t, boldx false)} = 0, \\ \frac{\partial boldu}{\partial t} false(t, boldx false) + {normal∇}_{normalT} ((), \frac{1}{2} false| boldu false(t, boldx false) {false|}^{2} + g h false(t, boldx false)) \\ + {f false(boldx false) + ζ false(t, boldx false)} boldn false(boldx false) prefix\times boldu false(t, boldx false) = 0 . \end{array}

…”

Section: The Hccs Solver: a Cubed‐sphere Approximation For The Spherimentioning

confidence: 99%

“…We have

\begin{matrix} {normal∇}_{normalT, normalΔ} frakturh false({bold-italics}_{i, j}^{k} false) ≃ {normal∇}_{normalT} h false({bold-italics}_{i, j}^{k} false), \\ {normal∇}_{normalT, normalΔ} \cdot frakturu false({bold-italics}_{i, j}^{k} false) ≃ {normal∇}_{normalT} \cdot boldu false({bold-italics}_{i, j}^{k} false), \\ {normal∇}_{normalT, normalΔ} prefix\times frakturu false({bold-italics}_{i, j}^{k} false) ≃ {normal∇}_{normalT} prefix\times boldu false({bold-italics}_{i, j}^{k} false) . \end{matrix}\}

Since () is a centred approximation, each approximate differential operator is centred as well. Details on the design and applications of the discrete operators () are reported in Croisille (2013) 2015 and Brachet and Croisille (2019). They are not repeated here.…”

Section: The Hccs Solver: a Cubed‐sphere Approximation For The Spherimentioning

confidence: 99%

“…In this section, the Hermitian Compact Scheme on the Cubed Sphere (HCCS) is summarized (Croisille 2013(Croisille , 2015Brachet 2018;Brachet and Croisille 2019). Consider the spherical SW equations: T , with h the height of the atmosphere and u the tangential wind velocity.…”

Section: The Hccs Solver: a Cubed-sphere Approximation For The Spherimentioning

confidence: 99%

“…The finite‐difference scheme considered here (HCCS), uses the equiangular cubed sphere (Ronchi et al ., 1996). The approximation principle for linear convection was introduced in Croisille (2013) and Croisille (2015). It can be seen as a generalisation to the spherical context of the high‐order compact schemes, widely used in Computational Aeroacoustics (CAA; Lele, 1992; Yee and Sjögreen, 2004).…”

Section: Introductionmentioning

confidence: 99%

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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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“…In this section, the Hermitian Compact Scheme on the Cubed Sphere (HCCS) is summarized (Croisille 2013, 2015; Brachet 2018; Brachet and Croisille 2019). Consider the spherical SW equations:

(SW) \{\begin{array}{l} \frac{\partial h^{⋆}}{\partial t} false(t, boldx false) + {normal∇}_{normalT} \cdot {h^{⋆} false(t, boldx false) boldu false(t, boldx false)} = 0, \\ \frac{\partial boldu}{\partial t} false(t, boldx false) + {normal∇}_{normalT} ((), \frac{1}{2} false| boldu false(t, boldx false) {false|}^{2} + g h false(t, boldx false)) \\ + {f false(boldx false) + ζ false(t, boldx false)} boldn false(boldx false) prefix\times boldu false(t, boldx false) = 0 . \end{array}

…”

Section: The Hccs Solver: a Cubed‐sphere Approximation For The Spherimentioning

confidence: 99%

“…We have

\begin{matrix} {normal∇}_{normalT, normalΔ} frakturh false({bold-italics}_{i, j}^{k} false) ≃ {normal∇}_{normalT} h false({bold-italics}_{i, j}^{k} false), \\ {normal∇}_{normalT, normalΔ} \cdot frakturu false({bold-italics}_{i, j}^{k} false) ≃ {normal∇}_{normalT} \cdot boldu false({bold-italics}_{i, j}^{k} false), \\ {normal∇}_{normalT, normalΔ} prefix\times frakturu false({bold-italics}_{i, j}^{k} false) ≃ {normal∇}_{normalT} prefix\times boldu false({bold-italics}_{i, j}^{k} false) . \end{matrix}\}

Section: The Hccs Solver: a Cubed‐sphere Approximation For The Spherimentioning

confidence: 99%

Section: The Hccs Solver: a Cubed-sphere Approximation For The Spherimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Brachet

Croisille

2020

Quart J Royal Meteoro Soc

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“…In this context, accurately evaluating averaged quantities over the sphere such as mass, momentum, energy or total vorticity is particularly important. This is in particular the case for the finite difference scheme introduced in [4,5]. This scheme uses discrete unknowns located at the nodes of the Cubed Sphere.…”

Section: Introductionmentioning

confidence: 99%

An efficient quadrature rule on the Cubed Sphere

Portelenelle

Croisille

2018

Journal of Computational and Applied Mathematics

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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 quadrature 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.

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A center compact scheme for the Shallow Water equations on the sphere

Brachet

Croisille

2022

Computers & Fluids

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Hermitian approximation of the spherical divergence on the Cubed-Sphere

Cited by 4 publications

References 25 publications

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

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

An efficient quadrature rule on the Cubed Sphere

A center compact scheme for the Shallow Water equations on the sphere

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