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 [A. Debussche, T. Dubois, and R. Temam, Theoret. Comput. Fluid Dyn., 7 (1995), pp. 279--315; M. Marion and R. Temam, SIAM J. Numer. Anal., 26 (1989), p...
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| description | This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier--Stokes equation on the sphere. It extends the work of [A. Debussche, T. Dubois, and R. Temam, Theoret. Comput. Fluid Dyn., 7 (1995), pp. 279--315; M. Marion and R. Temam, SIAM J. Numer. Anal., 26 (1989), pp. 1139--1157; J. Shen and R. Temam, Proceedings of the International Conference on Nonlinear Evolution Partial Differential Equations, AMS, Providence, RI, 1997, pp. 363--376] 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 pseudospectral 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. |
| doi_str_mv | 10.1137/040612567 |
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J ; FREEDEN, W</creator><creatorcontrib>FENGLER, M. J ; FREEDEN, W</creatorcontrib><description>This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier--Stokes equation on the sphere. It extends the work of [A. Debussche, T. Dubois, and R. Temam, Theoret. Comput. Fluid Dyn., 7 (1995), pp. 279--315; M. Marion and R. Temam, SIAM J. Numer. Anal., 26 (1989), pp. 1139--1157; J. Shen and R. Temam, Proceedings of the International Conference on Nonlinear Evolution Partial Differential Equations, AMS, Providence, RI, 1997, pp. 363--376] 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 pseudospectral 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.</description><identifier>ISSN: 1064-8275</identifier><identifier>EISSN: 1095-7197</identifier><identifier>DOI: 10.1137/040612567</identifier><identifier>CODEN: SJOCE3</identifier><language>eng</language><publisher>Philadelphia, PA: Society for Industrial and Applied Mathematics</publisher><subject>Algorithms ; Computational methods in fluid dynamics ; Euclidean space ; Exact sciences and technology ; Fluid dynamics ; Fundamental areas of phenomenology (including applications) ; Geometry ; Global analysis, analysis on manifolds ; Mathematical analysis ; Mathematics ; Meteorology ; Methods ; Navier-Stokes equations ; Partial differential equations ; Physics ; Sciences and techniques of general use ; Special functions ; Topology. Manifolds and cell complexes. 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J</creatorcontrib><creatorcontrib>FREEDEN, W</creatorcontrib><title>A nonlinear galerkin scheme involving vector and tensor spherical harmonics for solving the incompressible navier-stokes equation on the sphere</title><title>SIAM journal on scientific computing</title><description>This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier--Stokes equation on the sphere. It extends the work of [A. Debussche, T. Dubois, and R. Temam, Theoret. Comput. Fluid Dyn., 7 (1995), pp. 279--315; M. Marion and R. Temam, SIAM J. Numer. Anal., 26 (1989), pp. 1139--1157; J. Shen and R. Temam, Proceedings of the International Conference on Nonlinear Evolution Partial Differential Equations, AMS, Providence, RI, 1997, pp. 363--376] 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 pseudospectral 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. 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J ; FREEDEN, W</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c321t-662e12330603c13d1ed1094b8b7731c295e014cb8b691655c394f03445b4627d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2005</creationdate><topic>Algorithms</topic><topic>Computational methods in fluid dynamics</topic><topic>Euclidean space</topic><topic>Exact sciences and technology</topic><topic>Fluid dynamics</topic><topic>Fundamental areas of phenomenology (including applications)</topic><topic>Geometry</topic><topic>Global analysis, analysis on manifolds</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Meteorology</topic><topic>Methods</topic><topic>Navier-Stokes equations</topic><topic>Partial differential equations</topic><topic>Physics</topic><topic>Sciences and techniques of general use</topic><topic>Special functions</topic><topic>Topology. Manifolds and cell complexes. 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J</au><au>FREEDEN, W</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A nonlinear galerkin scheme involving vector and tensor spherical harmonics for solving the incompressible navier-stokes equation on the sphere</atitle><jtitle>SIAM journal on scientific computing</jtitle><date>2005-01-01</date><risdate>2005</risdate><volume>27</volume><issue>3</issue><spage>967</spage><epage>994</epage><pages>967-994</pages><issn>1064-8275</issn><eissn>1095-7197</eissn><coden>SJOCE3</coden><abstract>This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier--Stokes equation on the sphere. It extends the work of [A. Debussche, T. Dubois, and R. Temam, Theoret. Comput. Fluid Dyn., 7 (1995), pp. 279--315; M. Marion and R. Temam, SIAM J. Numer. Anal., 26 (1989), pp. 1139--1157; J. Shen and R. Temam, Proceedings of the International Conference on Nonlinear Evolution Partial Differential Equations, AMS, Providence, RI, 1997, pp. 363--376] 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 pseudospectral 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.</abstract><cop>Philadelphia, PA</cop><pub>Society for Industrial and Applied Mathematics</pub><doi>10.1137/040612567</doi><tpages>28</tpages><oa>free_for_read</oa></addata></record> |
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| subjects | Algorithms Computational methods in fluid dynamics Euclidean space Exact sciences and technology Fluid dynamics Fundamental areas of phenomenology (including applications) Geometry Global analysis, analysis on manifolds Mathematical analysis Mathematics Meteorology Methods Navier-Stokes equations Partial differential equations Physics Sciences and techniques of general use Special functions Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds |
| title | A nonlinear galerkin scheme involving vector and tensor spherical harmonics for solving the incompressible navier-stokes equation on the sphere |
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