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Fast Legendre spectral method for computing the perturbation of a gradient temperature field in an unbounded region due to the presence of two spheres
Author(s) -
Chowdhury A.,
Christov C. I.
Publication year - 2010
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20479
Subject(s) - legendre polynomials , mathematics , mathematical analysis , legendre function , spheres , uniqueness , taylor series , legendre transformation , exponential function , spherical harmonics , associated legendre polynomials , spectral element method , perturbation (astronomy) , transformation (genetics) , finite element method , mixed finite element method , physics , classical orthogonal polynomials , gegenbauer polynomials , quantum mechanics , astronomy , orthogonal polynomials , thermodynamics , biochemistry , chemistry , gene
The temperature distribution around two spheres is considered when the main field has a constant gradient at infinity. Bispherical coordinates are used, together with a transformation of the dependent variable that leads to separation of variables. Then the solution can be sought in Legendre series with respect to one of the bispherical coordinates. An important element of the proposed work is the effective way to reduce an essentially 3D problem to a set of three 2D problems. The Legendre spectral method is shown to have an exponential convergence which is confirmed by the computations. The efficiency is so high that even for the hard cases of two closely situated spheres, an accuracy of 10 −10 is achieved with as few as 20 terms in the expansion. Solutions with both longitudinal and transverse gradients at infinity are obtained, and the contour lines of the temperature field are presented graphically. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010