Renormalized Coordinate Stretching: A Generalization of Shoot and Fit with Application to Stellar Structure

Peter D. Usher

Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, USA.

DOI: 10.4236/ijaa.2013.33039                

Abstract

The standard shooting and fitting algorithm for non-linear two-point boundary value problems derives from conventional coordinate perturbation theory. We generalize the algorithm using the renormalized perturbation theory of strained coordinates. This allows for the introduction of an arbitrary function, which may be chosen to improve numerical convergence. An application to a problem in stellar structure exemplifies the algorithm and shows that, when used in conjunction with the standard procedure, it has superior convergence compared to the standard one alone.

Keywords

Singular Non-Linear Boundary Value Problems; Renormalized Coordinate Stretching; Poincaré-Lighthill Perturbation Theory; Numerical Analysis; Stellar Structure

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. B. Haselgrove and F. Hoyle, “A Mathematical Discussion of the Problem of Stellar Evolution, with Reference to the Use of an Automatic Digital Computer,” Monthly Notices of the Royal Astronomical Society, Vol. 116, 1956, pp. 515-526.
[2]	W. H., Press, S. A., Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes,” Cambridge University Press, Cambridge, 2007.
[3]	S. M. Roberts and J. S. Shipman, “Two-Point Boundary Value Problems: Shooting Methods,” Elsevier, New York, 1972.
[4]	A. Lindstedt, “Ueber Die Integration Einer Für Die Str?rungstheorie Wichtigen Differentialgleichung,” Astronomische Nachrichten, Vol. 103, No. 14, 1882, pp. 211-220.
[5]	H. Poincaré, “Les Méthodes Nouvelles de la Méchanique Céleste,” Gauthier-Villars, Paris, 1892.
[6]	M. J. Lighthill, “A Technique for Rendering Approximate Solutions to Physical Problems Uniformly Convergent,” Philosophical Magazine, Vol. 40, 1949, pp. 1179-1201.
[7]	A. S. Nayfeh, “Perturbation Methods,” Wiley, New York, 2000. doi:10.1002/9783527617609
[8]	J. Kevorkian and J. D. Cole, “Perturbation Methods in Applied Mathematics,” Springer Verlag, New York, 2010.
[9]	M. Pritulo, “On the Determination of Uniformly Accurate Solutions of Differential Equations by the Method of Perturbation of Coordinates,” Journal of Applied Mathematics and Mechanics, Vol. 26, No. 3, 1962, pp. 661-667. doi:10.1016/0021-8928(62)90034-5
[10]	P. D. Usher, “Some New Algorithms for Stellar Structure,” Ph.D. Thesis, Harvard University, Harvard, 1966.
[11]	S. Dai, “Poincaré-Lighthill-Kuo Method and Symbolic Computation,” Applied Mathematics and Mechanics, Vol. 22, No. 3, 2001, pp. 261-269. doi:10.1023/A:1015565502306
[12]	J. Reiter, R. Bulirsch and J. Pfleiderer, “A Multiple Shooting Approach for the Numerical Treatment of Stellar Structure and Evolution,” Astronomische Nachrichten, Vol. 315, No. 3, 1994, pp. 205-234. doi:10.1002/asna.2103150304
[13]	B. L. Smith, “Strained Coordinate Methods in Rotating Stars,” Astrophysics and Space Science, Vol. 47, No. 1, 1977, pp. 61-78.
[14]	M. Schwarzschild and R. H?rm, “Evolution of Very Massive Stars,” Astrophysical Journal, Vol. 128, 1958, pp. 348-360.
[15]	R. Stothers, “Evolution of O Stars. I. Hydrogen-Burning,” Astrophysical Journal, Vol. 138, 1963, pp. 1074-1084.
[16]	B. Maxfield, “Essential Mathcad,” Academic Press, Burlington, 2007.