The FARGO algorithm for the azimuthal transport turns out to be able to speed up by about an order of magnitude the numerical simulation of a differentially rotating disk, with a smaller numerical viscosity than the usual transport algorithm. It has been validated by many tests on the embedded protoplanet problem. It is worth mentioning that the FARGO transport algorithm must be used with a good understanding of the physical processes at work in the system. In particular, the timestep given by Eq. (15) must be short compared to all the physical time scales relevant for the system. In the case we have presented in this paper this is automatically ensured by the set of Eqs. (13) to (15), but if additional physics is to be added (magnetic field, radiative transfer, etc.), the timestep limit needs to be carefully worked out. Furthermore, no advantage is gained in using FARGO in problems where the perturbed velocity is comparable to the rest velocity. It is the case for instance of the gas flow in a galactic bar. This does not mean that the FARGO algorithm leads to wrong results in that case, but simply that it will not be better than a standard method, both in terms of numerical diffusivity and execution time. On the other hand, the FARGO algorithm appears to be very well suited to all the cases where the perturbed velocities in any differentially rotating disk are small compared to the unperturbed velocities, which does not mean that the problem under consideration has to be linear; indeed the relative perturbed amplitude can be arbitrarily high (see e.g.Sect. 5 in which the protoplanet wake generates shocks in the disk). More generally the FARGO algorithm can be used to describe the HD evolution of any sheared fluid on a fixed orthogonal eulerian grid.

I am indebted to R.P. Nelson for many valuable discussions and suggestions in the course of this work, as well as for a careful reading of the successive drafts of this paper, to J.M. Stone for advice on eulerian numerical simulations, to J.C.B. Papaloizou for many discussions on the protoplanet migration problem, to Willy Kley for advice on the corotating frame runs, and to an anonymous referee whose comments led to an improvement of the paper. This work was done in the research network "Accretion onto black holes, compact stars and protostars'' funded by the European Commission under contract number ERBFMRX-CT98-0195.

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