中国物理B ›› 2004, Vol. 13 ›› Issue (10): 1770-1776.doi: 10.1088/1009-1963/13/10/035

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Numerical solution of Helmholtz equation of barotropic atmosphere using wavelets

汪萍, 戴新刚   

  1. Key Laboratory of Regional Climate-Environment for Temperate East Asia, Institute of Atmospheric Physics, Chinese Academy of Sciences, START Regional Center for Temperate East Asia, Beijing 100029, China
  • 收稿日期:2004-02-18 修回日期:2004-05-20 出版日期:2004-10-20 发布日期:2005-06-20
  • 基金资助:
    Project supported by the National Key Planning Development for Basic Research (Grant No 1999043403) and the National Natural Science Foundation of China (Grant No 49875024).

Numerical solution of Helmholtz equation of barotropic atmosphere using wavelets

Wang Ping (汪萍), Dai Xin-Gang (戴新刚)   

  1. Key Laboratory of Regional Climate-Environment for Temperate East Asia, Institute of Atmospheric Physics, Chinese Academy of Sciences, START Regional Center for Temperate East Asia, Beijing 100029, China
  • Received:2004-02-18 Revised:2004-05-20 Online:2004-10-20 Published:2005-06-20
  • Supported by:
    Project supported by the National Key Planning Development for Basic Research (Grant No 1999043403) and the National Natural Science Foundation of China (Grant No 49875024).

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摘要： The numerical solution of the Helmholtz equation for barotropic atmosphere is estimated by use of the wavelet-Galerkin method. The solution involves the decomposition of a circulant matrix consisting up of 2-term connection coefficients of wavelet scaling functions. Three matrix decompositions, i.e. fast Fourier transformation (FFT), Jacobian and QR decomposition methods, are tested numerically. The Jacobian method has the smallest matrix-reconstruction error with the best orthogonality while the FFT method causes the biggest errors. Numerical result reveals that the numerical solution of the equation is very sensitive to the decomposition methods, and the QR and Jacobian decomposition methods, whose errors are of the order of 10^{-3}, much smaller than that with the FFT method, are more suitable to the numerical solution of the equation. With the two methods the solutions are also proved to have much higher accuracy than the iteration solution with the finite difference approximation. In addition, the wavelet numerical method is very useful for the solution of a climate model in low resolution. The solution accuracy of the equation may significantly increase with the order of Daubechies wavelet.

关键词: wavelet-Galerkin method, Helmholtz equation, matrix decomposition, barotropic atmosphere

Abstract: The numerical solution of the Helmholtz equation for barotropic atmosphere is estimated by use of the wavelet-Galerkin method. The solution involves the decomposition of a circulant matrix consisting up of 2-term connection coefficients of wavelet scaling functions. Three matrix decompositions, i.e. fast Fourier transformation (FFT), Jacobian and QR decomposition methods, are tested numerically. The Jacobian method has the smallest matrix-reconstruction error with the best orthogonality while the FFT method causes the biggest errors. Numerical result reveals that the numerical solution of the equation is very sensitive to the decomposition methods, and the QR and Jacobian decomposition methods, whose errors are of the order of $10^{-3}$, much smaller than that with the FFT method, are more suitable to the numerical solution of the equation. With the two methods the solutions are also proved to have much higher accuracy than the iteration solution with the finite difference approximation. In addition, the wavelet numerical method is very useful for the solution of a climate model in low resolution. The solution accuracy of the equation may significantly increase with the order of Daubechies wavelet.

Key words: wavelet-Galerkin method, Helmholtz equation, matrix decomposition, barotropic atmosphere

中图分类号:  (Finite-element and Galerkin methods)

  • 02.70.Dh
02.30.Jr (Partial differential equations) 02.60.Lj (Ordinary and partial differential equations; boundary value problems) 02.10.Yn (Matrix theory) 92.60.-e (Properties and dynamics of the atmosphere; meteorology)