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Advances in Atmospheric Sciences

, Volume 33, Issue 3, pp 377–390 | Cite as

A global spectral element model for poisson equations and advective flow over a sphere

  • Huan MeiEmail author
  • Faming Wang
  • Zhong Zeng
  • Zhouhua Qiu
  • Linmao Yin
  • Liang Li
Article

Abstract

A global spherical Fourier–Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre–Gauss–Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity, with only one element. Then, the partial differential equations are solved on the longitude–latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. Highprecision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.

Key words

spectral element method spherical coordinates Poisson equations advective equation Legendre–Gauss–Radau 

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© Institute of Atmospheric Physics/Chinese Academy of Sciences, and Science Press and Springer-Verlag Berlin Heidelberg 2016

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Authors and Affiliations

  • Huan Mei
    • 1
    Email author
  • Faming Wang
    • 1
  • Zhong Zeng
    • 2
  • Zhouhua Qiu
    • 2
  • Linmao Yin
    • 2
  • Liang Li
    • 3
  1. 1.Institute of OceanologyChinese Academy of SciencesQingdaoChina
  2. 2.Department of Engineering Mechanics, College of Aerospace EngineeringChongqing UniversityChongqingChina
  3. 3.Department of Chemical and Biological EngineeringChalmers University of TechnologyGothenburgSweden

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