Towards a new generation of multi-dimensional stellar evolution models: development of an implicit hydrodynamic code

This paper describes the first steps in the development of a new multi-dimensional time implicit code devoted to the study of hydrodynamical processes in stellar interiors. The code solves the hydrodynamical equations in spherical geometry and is based on the finite volume method. Radiation transport is taken into account within the diffusion approximation. Realistic equation of state and opacities are implemented, allowing study of a wide range of the problems characteristic of stellar interiors. We describe the numerical method and various standard tests performed to validate the method in detail. We present preliminary results devoted to describing stellar convection. We first performed a local simulation of convection in the surface layers of a A-type star model. This simulation tested the ability of the code to address stellar conditions and to validate our results, since they can be compared to similar previous simulations based on explicit codes. We then present a global simulation of turbulent convective motions in a cold giant envelope, covering 80% in radius of the stellar structure. Although our implicit scheme is unconditionally stable, we show that in practice there is a limitation on the time step that prevents the flow moving over several cells during a time step. Nevertheless, in the cold giant model we reach a hydro CFL number of 100. We also show that we are able to address flows with a wide range of Mach numbers (10(-3) less than or similar to M-s less than or similar to 0.5), which is impossible with an anelastic approach. Our first developments are meant to demonstrate that applying an implicit scheme to a stellar evolution context is perfectly thinkable and to provide useful guidelines for optimising the development of an implicit multi-dimensional hydrodynamical code.