## 2. Numerical methodsWe investigate the evolution of disks exposed to an external radiation field as shown in Fig. 1, using a modified version of the 2D radiation hydrodynamics code described in Yorke & Welz (1996, Paper I) and Richling & Yorke (1997, Paper II). In addition to the usual hydrodynamics with heating and cooling this code calculates the direct hydrogen-ionizing stellar radiation field as well as the diffuse radiation fields resulting from recombination of hydrogen into the ground state and from scattering on dust grains. The diffuse radiation fields are treated in the flux-limited diffusion (FLD) approximation (Levermore & Pomraning 1981) as described and implemented by Yorke & Kaisig (1995) for multiply nested grids. We also solve the Poisson equation for the gas' self-gravity and the FLD equations for grey (IR) continuum radiation transfer in order to determine the dust temperature (see Yorke et al. 1995). The code described in Paper I and II uses cylindrical coordinates and solves the equations in the quadrant with and assuming axial symmetry and symmetry with respect to the equatorial plane. This code design is well suited for previous work where the ionizing source was located at the center of the disk.
Moving the radiation source along the rotation axis out of the computational domain breaks the symmetry with respect to the equatorial plane, necessitating several modifications to the code. The numerical grid was extended to include the quadrant with and . The nested grids are now distributed as shown in Fig. 2 for an example of two grids. This arrangement maintains the enhanced resolution for the inner regions of the disk where most of the mass is concentrated. We also changed the boundary conditions at the lower boundary of the grid. For the coarsest grid we now assume semi-permeable outflow boundary conditions. All inner grids get their boundary conditions from the next coarser grid. Almost all parts of the code were affected by this procedure. A detailed description of the method of explicit nested grids and the treatment of grid interaction can be found in Yorke & Kaisig (1995).
Major changes were necessary for the definition of the lines of
sight across the numerical grid for the integration of the direct
EUV radiation. In Papers I and II these lines of sight
extended radially from the central source located at
to the centers of the outermost The coupling between the grid and the lines of sight is treated in the same way as in the original code. The magnification on the right of Fig. 2 displays the position of the physical variables on the grid. Scalar variables are located at the center of a cell, but the optical depth is defined at the intersection of the lines of sight with the boundaries of the cells. Therefore a mean optical depth is calculated from where we use the ray segments as weight factors. The summation extends over all lines of sight crossing the cell. From we get the photon density within the cell. is the stellar
EUV photon rate and which includes both extinction resulting from ionization of
hydrogen and dust extinction. Here Outside the computational domain we assume constant density (). Furthermore, we assume that the gas is fully ionized outside the grid (). Thus, only dust extinction is effective when calculating the optical depth of each line of sight at the upper boundary of the coarsest grid where is the ray segment between the star and the upper boundary and . In our simulations the outer ray segments are of order , which leads to a slight reduction of the EUV photon flux at the upper boundary of the grid. This simple assumption for the upper boundary condition is an attempt to take at least some extinction outside the grid into account. A proper treatment would require a more realistic density distribution towards the ionizing star and the integration of the optical depth along . At the interface to the next finer grid is stored for use as the boundary condition when calculating on the finer grid. Note that the more general situation - an off-axis location of the source - is only possible using a 3D code. © European Southern Observatory (ESO) 1998 Online publication: November 9, 1998 |