Tensor calculus in polar coordinates using Jacobi polynomials

JOURNAL OF COMPUTATIONAL PHYSICS

卷:325

Article

Vasil, Geoffrey M.¹  Burns, Keaton J.²  Lecoanet, Daniel^3,4  Olver, Sheehan¹  Brown, Benjamin P.^5,6  Oishi, Jeffrey S.⁷ 

[1] Univ Sydney, Sch Math & Stat, Sydney, NSW 2006, Australia

[2] MIT, Dept Phys, Cambridge, MA 02139 USA

[3] Univ Calif Berkeley, Dept Phys, Berkeley, CA 94720 USA

[4] Univ Calif Berkeley, Theoret Astrophys Ctr, Berkeley, CA 94720 USA

[5] Univ Colorado, LASP, Boulder, CO 80309 USA

[6] Univ Colorado, Dept Astrophys & Planetary Sci, Boulder, CO 80309 USA

[7] Bates Coll, Dept Phys & Astrophys, Lewiston, ME 04240 USA

关键词: Numerical analysis;    Partial differential equations;    Orthogonal polynomials;    Jacobi polynomials;    Fluid mechanics;    Pipe flow;   

DOI  :  10.1016/j.jcp.2016.08.013

来源: Elsevier

【 摘 要 】

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free