1 Introduction

Due to extensive observational activities in the last decade we dispose of a huge amount of accurate experimental data about pulsating stars. In particular the microlensing projects OGLE (e.g. Kaluzny et al. 1998) and MACHO (e.g. Alcock et al. 1996, 1998) as by-products provide a large number of high-quality photometric data of radially pulsating stars (e.g. Cepheids, RR Lyrae, W Virginis, AGB stars), many of them newly discovered. From the theoretical point of view these data in a unique way constrain our theoretical models. We use this models to understand the physical processes acting in pulsating stars. Consequently, in order to benefit from the variety of detailed observations, high-quality computations of stellar pulsations are necessary in order to reproduce the observational facts with the required accuracy.

The aim of this paper, which is the first in a series dealing with nonlinear pulsation models, is to review the fundamental physical and numerical methods we employ to compute realistic models of radially oscillating stars. The philosophy behind the development of this model is twofold. On the one hand it is essential to consider all physical processes expected to be important in a pulsating envelope. Apart from fluid dynamics this encloses time-dependent radiative transfer and time-dependent turbulent convection. Consequently a comprehensive description of stellar oscillations requires a "convective radiation hydrodynamical" picture. By that we mean that the nonlinear equations of radiation hydrodynamics including a model for convective energy transfer have to be solved for the case of radial stellar pulsations.

The numerical difficulties usually encountered when investigating such complex nonlinear systems under astrophysical conditions naturally lead to our second demand, namely to use appropriate numerical methods. Analogous to most astrophysical problems in fluid dynamics, stellar pulsation flows exhibit a number of complicated nonlinear features like shock waves or moving ionization fronts, often acting on a non-trivial hydrostatic stellar background structure. In this context the main numerical problem is the sufficient resolution of all important structures during the whole computation. The basic consequence is to use an adaptive grid algorithm which is able to track all important flow features continuously with high spatial resolution. This involves a number of implications for the formulation of the physical equations and the numerical methods adopted for solving the discretized system of convective radiation hydrodynamics.

The paper is divided into two main parts: Sect. 2 summarizes all physical ingredients of the model and Sect. 3 is devoted to the numerical method of solution. Section 4 closes the paper with some general remarks about the applicability of the model.