6 The influence of the spatial grid

Let us next investigate how the evolution of structure may be affected by the spatial mesh used in the simulations. We compare the reference grid (Sect. 4) with two modified grids: one with double the original grid spacing and one with a grid spacing increasing linearly as a function of radius, as used e.g. by Feldmeier et al. (1997b). As we are interested in the effect of grid spacing on the outer wind evolution, we keep the grid below 10 R_*unchanged. To avoid a discontinuity in the mesh spacing, the doubling is implemented over a transition region between 10 and 12 R_*, where $\Delta r$ increases linearly from 0.01 to 0.02 R_*. For the second grid, the grid spacing increases from 0.01 R_* at 11 R_* to 0.1 R_* at $R_{\rm max}$. For both models, the time averages were again calculated over a 500 ksec interval after 2 Msec to allow the response to the initial condition to die away.

In Fig. 7 we show the influence of the spatial grid on the clumping factor and velocity dispersion. The solid line corresponds to the reference grid, the dotted line to the grid with increasing step size, and the dashed line to the grid with doubled spacing. The most obvious difference is in the clumping factor. The strongest clumps in the reference model occur between 10 and 15 R_* as a result of collisions and are often as narrow as 0.05 R_*. These clumps are smeared by the coarser grids. The agreement in the inner wind is much better, of course, but even there there are some minor differences, due to the backscattering effect mentioned previously.

The instability of line driving generates structure down to the smallest spatial scales of our radial mesh. In this sense, hydrodynamical simulations of line-driven flows are never truly grid-independent, as the grid sets an artificial limit to the resolution scale. As the instability growth rate is strongest for variations near and below the Sobolev length, the grid in the inner wind is typically chosen to resolve some fraction of this length (OCR). The results in this section indicate that for structure generated on a given inner wind grid, care must be taken in increasing the grid spacing in the outer wind, as features can remain surprisingly narrow up to quite large distances when adequately resolved.