Analysis of Methods for Computing the Trajectories of Dust Particles in a Gas–Dust Circumstellar Disk

Abstract

A systematic analysis of methods for computing the trajectories of solid-phase particles applied in modern astrophysics codes designed for modeling gas-dust circumstellar disks has been carried out for the first time. The motion of grains whose velocities are determined mainly by the gas drag, that is, for which the stopping time or relaxation time for the velocity of the dust to the velocity of the gas tstop is less than or comparable to the rotation period, are considered. The methods are analyzed from the point of view of their suitability for computing the motions of small bodies, including dust grains less than 1 µm in size, which are strongly coupled to the gas. Two test problems are with analytical solutions. Fast first order accurate methods that make it possible to avoid additional restrictions on the time step size τ due to gas drag in computations of the motion of grains of any size are presented. For the conditions of a circumstellar disk, the error in the velocity computations obtained when using some stable methods becomes unacceptably large when the time step size is τ > tstop. For the radial migration of bodies that exhibit drifts along nearly Keplerian orbits, an asymptotic approximation, sometimes called the short friction time approximation or drift flux model, gives a relative error for the radial-velocity computations equals to St 2 where St is the Stokes number, the ratio of the stopping time of the body to some fraction of the rotation period (dynamical time scale) in the disk.

Small dust grain dynamics on adaptive mesh refinement grids

“…They also require extremely high spatial resolu-tion in order to capture correctly the dephasing between gas and dust, i.e., the dissipation of energy through the drag . Some efficient bifluid treatments were developed to cope with these issues (e.g., semi implicit methods, Lorén-Aguilar & Bate 2014Bate , 2015Stoyanovskaya et al 2017Stoyanovskaya et al , 2018. As an alternative way to model correctly these mixtures, Laibe & Price (2014a) proposed an algorithm for smoothed Particle Hydrodynamic codes (SPH, Lucy 1977;Gingold & Monaghan 1977) using a monofluid formalism.…”

Analysis of Numerical Algorithms for Computing Rapid Momentum Transfers between the Gas and Dust in Simulations of Circumstellar Disks

“…This scheme is constructed by analogy with the semi-implicit scheme, or mixed-time-layer scheme, of [8,45], in which stability of the solution is provided by using the aerodynamical velocity drag from the following time layer in the computation.…”

Development and application of fast methods for computing momentum transfer between gas and dust in supercomputer simulation of planet formation

“…A review of the literature on the methods for computing the gas-dust medium in circumstellar discs can be found e.g. in [2].…”

“…Thus, for dust particles with the size of about 1 µm, the stopping time t stop in the disc is about 100 s (see e.g. [3,4]), whereas the disc dynamics should be simulated for 10 4 years and longer. It means that the numerical solution of non-stationary equations for a multiphase gas-dust medium at times of disc dynamics using explicit integration schemes will require about 10 10 time steps because each time step τ should satisfy the condition τ < t stop .…”

“…Here, we compare the earlier proposed grid [2,5] and free Lagrangian methods [6,7,8] for solving the complete problem of momentum transfer between gas and dust in a circumstellar disc. We demonstrate that both the Euler and the Lagrangian methods, in which the momentum conservation law is strictly fulfilled locally and globally, make it possible to obtain an acceptable accuracy of solution even if conditions (1) and (2) are violated.…”