A multiple shooting approach for the numerical treatment of stellar structure and evolution

Abstract

AbstractWe present a new numerical method for solving the system of partial differential equations describing the structure and evolution of a spherically symmetric star. As usual, we employ the transversal method of lines in order to split the equations into a coupled spatial and temporal part. The novel features of the algorithm are the following: (a) Instead of using the Lagrangian picture we formulate the system of partial differential equations in the Eulerian picture. (b) We reformulate the equations of stellar structure as a multipoint boundary‐value problem. By means of this reformulation the rather clumsy iterative matching procedure of stellar atmosphere and interior is avoided. (c) The multipoint boundary‐value problem is solved by the multiple shooting method. This approach not only ensures a high accuracy of the stellar models calculated at each time step but also allows the free boundaries inside the star due to different energy transport mechanisms to be located exactly. (d) The time derivatives involved in the stellar‐structure equations are discretized implicitly to second order accuracy. Moreover, at each time step, the chemical abundances are determined by using a sophisticated update procedure. In this way, a high accuracy is achieved with respect to the integration in time.The algorithm has turned out to be exceedingly reliable and numerically accurate. This is shown by the evolution of a 1 M⊙ star up to the hydrogen‐shell burning phase. In this example, the virial theorem, the law of mass conservation, and the law of energy conservation is fulfilled to a hitherto unattainable degree of accuracy.Since the multiple shooting method, which is at the heart of our approach, is a perfect example of a parallel algorithm, the computational speed of the algorithm might be substantially improved provided easy‐to‐program, high‐performance parallel computers with sufficiently many processors become available in the near future.