1 Introduction

We want hereafter to model the hydrodynamical (HD) evolution of a disk described on a fixed polar eulerian grid. For the sake of simplicity we are only going to deal with a two dimensional Keplerian disk, but the algorithm can be extended with little additional effort to any gaseous thin or thick disk in differential rotation. Usually in this kind of numerical simulations the timestep is limited by the Courant Friedrich Levy (CFL) condition at the inner boundary, where the motion is fast and the cells are narrow. Indeed, the ratio of the distance swept by the material in one timestep to the cell width must be lower than unity over the whole grid, otherwise a numerical instability occurs (i.e. non physical short-wavelength oscillations appear, grow exponentially and spoil the model). In a Keplerian disk this ratio (which we call hereafter the CFL ratio) decreases as r^-3/2. Since in most cases the "interesting region'' of the grid is located much further than the grid inner boundary, the CFL ratio in the region of interest is much smaller than unity, which corresponds to a waste of computing time, and, as we are going to see below, to an enhanced undesirable numerical viscosity. The most obvious solution to get rid of such a limitation is to work in the comoving frame. Unfortunately, most finite-difference HD eulerian codes require an orthogonal system of coordinates (Stone & Norman 1992), which makes them unsuitable if one wants to work in the comoving frame in a differentially rotating disk, and even a non-orthogonal grid eulerian code would be unable to track accurately the fluid motion after a few orbits, due to the strong winding of the coordinate system. On the other hand, one can adopt a Lagrangian description of the disk (Whitehurst 1995), but the implementation is much more tricky and difficult. Furthermore, the geometry of an accretion disk provides a polar mesh as a natural grid. We describe hereafter a simple method which enables one to work on a fixed polar grid and to get rid of the CFL condition on the average azimuthal velocity at each radius.