Three ways to solve the Poisson equation on a sphere with Gaussian forcing

Boyd, John P.; Zhou, Cheng

doi:10.1016/j.jcp.2009.03.023

Cited by 4 publications

(5 citation statements)

References 41 publications

(37 reference statements)

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“…Recently, Boyd and Zhou [17] filled this gap by discovering an analytical solution for the Poisson equation when forced by a single Gaussian, which we may dub the "P-function", P. The Poisson equation can then be solved for a general inhomogeneous term by expanding the forcing (vorticity) as a series of radial basis functions; the Poisson solution for the stream function is then a sum of translated copies of P, each multiplied by the RBF coefficient of the vorticity series. This advance has made it feasible to extend vortex methods on the sphere to Gaussian blobs instead of point vortices.…”

Section: Point Vortices Gaussian Blobs and Quasi-interpolationmentioning

confidence: 98%

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RBF-vortex methods for the barotropic vorticity equation on a sphere

Xiao

Wang

Boyd

2015

Journal of Computational Physics

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Section: Point Vortices Gaussian Blobs and Quasi-interpolationmentioning

confidence: 98%

“…Note that this algorithm does not exploit the analytical Poisson solution of Boyd and Zhou [17]. Preprocessing steps:…”

Section: The Lagrangian Vortex/rbf Algorithmmentioning

confidence: 99%

RBF-vortex methods for the barotropic vorticity equation on a sphere

Xiao

Wang

Boyd

2015

Journal of Computational Physics

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“…Sakajo [24] has developed a fast tree-code reducing the O(N 2 ) computational cost to O(N log 3 N) for N interacting point vortices. Regularized approximations to the delta distributions provide more accurate representations of continuous vorticity fields, but their solutions are no longer exact, as the kernel itself ought to deform due to shearing [25,26].…”

Section: Motivationmentioning

confidence: 99%

“…, n. The mapping I p q kℓ is the same as in (23). Note that the definition of Φ LT 0 in ( 21) is in fact equivalent to (25) for q = 1.…”

Section: Interactions Within Leaguesmentioning

confidence: 99%

Explicit, parallel Poisson integration of point vortices on the sphere

Myerscough

Frank

2016

Journal of Computational and Applied Mathematics

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Point vortex models are frequently encountered in conceptual studies in geophysical fluid dynamics, but also in practical applications, for instance, in aeronautics. In spherical geometry, the motion of vortex centres is governed by a dynamical system with a known Poisson structure. We construct Poisson integration methods for these dynamics by splitting the Hamiltonian into its constituent vortex pair terms. From backward error analysis, the method is formally known to provide solutions to a modified Poisson system with the correct bracket, but with a modified Hamiltonian function. Different orderings of the pairwise interactions are considered and also used for the construction of higher order methods. The energy and momentum conservation of the splitting schemes is demonstrated for several test cases. For particular orderings of the pairwise interactions, the schemes allow scalable parallelization. This results in a linear -as opposed to quadratic -scaling of computation time with system size when scaling the number of processors accordingly.

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“…However, numerical solutions may always be attained with the help of some specific methods. Different numerical techniques to solve Poisson's equation to obtain electrostatic potentials are found in the literature of Computational Physics and Applied Mathematics, which use Fourier series [1], approximate analytic solutions [2] and finite difference discretization scheme [3]. In many situations, to simplify the problem, one assumes a homogeneous medium and the absence of * Correspondence email address: asilva@nuclear.ufrj.br electric charges, which reduces the problem to solving the Laplace equation.…”

Section: Introductionmentioning

confidence: 99%

A numerical procedure to solve Poisson’s equation in spherical coordinates

Silva

Helayël-Neto

Costa

2021

Rev. Bras. Ensino Fís.

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This paper sets out to present a numerical procedure that solves Poisson's equation in a spherical coordinate system. To discretize this equation, integration techniques at the interfaces between different regions have been carried out allowing the calculation of both the potential and the corresponding field inside and outside a charge distribution. The Gauss-Seidel method is adopted to determine the potential in each region and the results, whenever compared with the analytical solutions found in the literature, come out very satisfactory, with errors less than 1% for distances of the order of 1 × 10 −14 m and, for larger distances, they never reach 4%.

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Three ways to solve the Poisson equation on a sphere with Gaussian forcing

Cited by 4 publications

References 41 publications

RBF-vortex methods for the barotropic vorticity equation on a sphere

RBF-vortex methods for the barotropic vorticity equation on a sphere

Explicit, parallel Poisson integration of point vortices on the sphere

A numerical procedure to solve Poisson’s equation in spherical coordinates

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