3 Hydrodynamic codes

The calculations presented here were performed with the three dimensional MHD code NIRVANA (here adapted to 2D) that has been described in depth elsewhere (Ziegler & Yorke 1997). Viscous forces have been added as described by Kley (1998).

In each case the equations are solved using a finite difference scheme on a discretised computational domain containing $N_{\rm r} \times N_{\phi }$grid cells, where the grid spacing in the $(r,\varphi)$ coordinates is uniform.

For the calculations presented here and listed in Table 1, no accretion onto the companion was allowed and three different resolutions have been used. The run with q=10^-3, listed as N1 in Table 1 used $N_{\rm r}=130$ and $N_{\phi}=384$. The other lower resolution runs used $N_{\rm r}=200$ and $N_{\phi}=200.$ One higher resolution run used $N_{\rm r}=400$ and $N_{\phi}=400$. The numerical method is based on the monotonic transport algorithm of Van Leer (1977), leading to the global conservation of mass and angular momentum. The evolution of the companion orbit was computed using a standard leapfrog integrator.

 

 

Table 1: This table lists the label given to each simulation, the resolution used for each calculation ( $N_{\rm r} \times N_{\phi }$), the companion-star mass ratio, and whether strong eccentricity growth was observed

Run	Resolution	q	e
			growth?
N1	$130 \times 384$	10-3	No
N2	$200 \times 200$	0.01	Yes
N3	$200 \times 200$	0.02	Yes
N4	$200 \times 200$	0.025	Yes
N5	$200 \times 200$	0.03	Yes
N6	$400 \times 400$	0.03	Yes

NIRVANA has been applied to a number of different problems including that of an accreting protoplanet embedded in a protostellar disc (NPMK). It was found to give results that are very similar to those obtained with other finite difference codes including FARGO (described in Masset 2000) and RH2D (described in Kley 1999).

In order to establish the reliability of the numerical results, simulations were also performed using an alternative code based on the FARGO fast advection method (see Masset 2000, NPMK). In this scheme, which is able to operate with longer time steps than NIRVANA, a fourth-order Runge Kutta scheme was used as orbit integrator. Results obtained with the two codes were very similar.