Numerical Treatment of Linear and Nonlinear Stellar Pulsations

Abstract

The linear stability analysis of stellar models poses a linear fourth or sixth order boundary eigenvalue problem. Methods for its numerical solution are reviewed, most of which face severe problems, if the ratio of the thermal and dynamical timescale falls below unity for a significant fraction of the stellar envelope considered. The extremely robust and highly accurate Riccati method is introduced and shown to be applicable to stellar stability problems with success even in these cases of strong deviations from adiabaticity. Numerical simulations of the evolution of a stellar instability into the nonlinear regime are still restricted to spherical geometry. We address the basic requirements for and problems connected with the simulation of radial pulsations. How violent artificial initial perturbations may be avoided and the extremely high accuracy requirements posed by the differences between the various energy forms can be met by strictly conservative numerical schemes is discussed.