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
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.