A Fast Parallel Simulation Code for Interaction between Proto-Planetary Disk and Embedded Proto-Planets: Implementation for 3D Code

Abstract

We develop a 3D simulation code for interaction between the proto-planetary disk and embedded proto-planets. The protoplanetary disk is treated as a three-dimensional (3D), self-gravitating gas whose motion is described by the locally isothermal Navier-Stokes equations in a spherical coordinate centered on the star. The differential equations for the disk are similar to those given in Kley et al. (2009) with a different gravitational potential that is defined in Nelson et al. (2000). The equations are solved by directional split Godunov method for the inviscid Euler equations plus operator-split method for the viscous source terms. We use a sub-cycling technique for the azimuthal sweep to alleviate the time step restriction. We also extend the FARGO scheme of Masset (2000) and modified in Li et al. (2001) to our 3D code to accelerate the transport in the azimuthal direction. Furthermore, we have implemented a reduced 2D (r, {theta}) and a fully 3D self-gravity solver on our uniform disk grid, which extends our 2D method (Li, Buoni, & Li 2008) to 3D. This solver uses a mode cut-off strategy and combines FFT in the azimuthal direction and direct summation in the radial and meridional direction. An initial axis-symmetric equilibrium disk is generated via iteration between the disk density profile and the 2D disk-self-gravity. We do not need any softening in the disk self-gravity calculation as we have used a shifted grid method (Li et al. 2008) to calculate the potential. The motion of the planet is limited on the mid-plane and the equations are the same as given in D'Angelo et al. (2005), which we adapted to the polar coordinates with a fourth-order Runge-Kutta solver. The disk gravitational force on the planet is assumed to evolve linearly with time between two hydrodynamics time steps. The Planetary potential acting on the disk is calculated accurately with a small softening given by a cubic-spline form (Kley et al. 2009). Since the torque is extremely sensitive to the position of the planet, we adopt the corotating frame that allows the planet moving only in radial direction if only one planet is present. This code has been extensively tested on a number of problems. For the earthmass planet with constant aspect ratio h = 0.05, the torque calculated using our code matches quite well with the the 3D linear theory results by Tanaka et al. (2002). The code is fully parallelized via message-passing interface (MPI) and has very high parallel efficiency. Several numerical examples for both fixed planet and moving planet are provided to demonstrate the efficacy of the numerical method and code.