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.

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