A MULTIPLE SHOOTING APPROACH FOR THE NUMERICAL TREATMENT OF STELLAR STRUCTURE AND EVOLUTION

Citation
J. Reiter et al., A MULTIPLE SHOOTING APPROACH FOR THE NUMERICAL TREATMENT OF STELLAR STRUCTURE AND EVOLUTION, Astronomische Nachrichten, 315(3), 1994, pp. 205-234
Citations number
71
Categorie Soggetti
Astronomy & Astrophysics
Journal title
ISSN journal
00046337
Volume
315
Issue
3
Year of publication
1994
Pages
205 - 234
Database
ISI
SICI code
0004-6337(1994)315:3<205:AMSAFT>2.0.ZU;2-K
Abstract
We present a new numerical method for solving the system of partial di fferential equations describing the structure and evolution of a spher ically symmetric star. As usual, we employ the transversal method of l ines in order to split the equations into a coupled spatial and tempor al part. The novel features of the algorithm are the following: (a) In stead of using the Lagrangian picture we formulate the system of parti al differential equations in the Eulerian picture. (b) We reformulate the equations of stellar structure as a multipoint boundary-value prob lem. By means of this reformulation the rather clumsy iterative matchi ng procedure of stellar atmosphere and interior is avoided. (c) The mu ltipoint boundary-value problem is solved by the multiple shooting met hod. This approach not only ensures a high accuracy of the stellar mod els calculated at each time step but also allows the free boundaries i nside the star due to different energy transport mechanisms to be loca ted exactly. (d) The time derivatives involved in the stellar-structur e equations are discretized implicitly to second order accuracy. Moreo ver, at each time step, the chemical abundances are determined by usin g a sophisticated update procedure. In this way, a high accuracy is ac hieved with respect to the integration in time. The algorithm has turn ed out to be exceedingly reliable and numerically accurate. This is sh own by the evolution of a 1 M. star up to the hydrogen-shell burning p hase. In this example, the virial theorem, the law of mass conservatio n, and the law of energy conservation is fulfilled to a hitherto unatt ainable degree of accuracy. Since the multiple shooting method, which is at the heart of our approach, is a perfect example of a parallel al gorithm, the computational speed of the algorithm might be substantial ly improved provided easy-to-program, high-performance parallel comput ers with sufficiently many processors become available in the near fut ure.