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A Nonlinear Galerkin Scheme Involving Vector and Tensor Spherical Harmonics for Solving the Incompressible Navier-Stokes Equation on the Sphere

  • This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier-Stokes equation on the sphere. It extends the work of Debussche, Marion,Shen, Temam et al. from one-dimensional or toroidal domains to the spherical geometry. In the first part, the method based on type 3 vector spherical harmonics is introduced and convergence is indicated. Further it is shown that the occurring coupling terms involving three vector spherical harmonics can be expressed algebraically in terms of Wigner-3j coefficients. To improve the numerical efficiency and economy we introduce an FFT based pseudo spectral algorithm for computing the Fourier coefficients of the nonlinear advection term. The resulting method scales with O(N^3), if N denotes the maximal spherical harmonic degree. The latter is demonstrated in an extensive numerical example.

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Author:Martin J. Fengler, Willi Freeden
Serie (Series number):Schriften zur Funktionalanalysis und Geomathematik (11)
Document Type:Working Paper
Language of publication:English
Year of Completion:2004
Year of Publication:2004
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2004/08/10
Tag:Inkompressibel Navier-Stokes; Nichtlineares Galerkinverfahren
Fast Pseudo Spectral Algorithm; Incompressible Navier-Stokes; Nonlinear Galerkin Method; Tensor Spherical Harmonics; Vector Spherical Harmonics
GND-Keyword:Navier-Stokes-Gleichung; Galerkin-Methode; Kugelflächenfunktion; Schnelle Fourier-Transformation; Globale nichtlineare Analysis; Kugel
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011