Two-fluid dusty gas in smoothed particle hydrodynamics: Fast and implicit algorithm for stiff linear drag

Abstract

Simulation of the dynamics of dust-gas circumstellar discs is crucial in understanding the mechanisms of planet formation. The dynamics of small grains in the disc is stiffly coupled to the gas, while the dynamics of grown solids is decoupled. Moreover, in some parts of the disc the concentration of the dust is low (dust to gas mass ratio is about 0.01), while in other parts it can be much higher. These factors place high requirements on the numerical methods for disc simulations. In particular, when gas and dust are simulated with two different fluids, explicit methods require very small timestep (must be less than dust stopping time t stop during which the velocity of a solid particle is equalized with respect to the gas velocity) to obtain solution, while some implicit methods requires high temporal resolution to obtain acceptable accuracy. Moreover, recent studies underlined that for Smoothed particle hydrodynamics (SPH) when the gas and the dust are simulated with different sets of particles only high spatial resolution h < c s t stop guaranties suppression of numerical overdissipation due to gas and dust interaction.To address these problems, we developed a fast algorithm based on the ideas of (1) implicit integration of linear (Epstein) drag and (2) exact conservation of local linear momentum. We derived formulas for monodisperse dust-gas in two-fluid SPH and tested the new method on problems with known analytical solutions. We found that our method is a promising alternative for the previously developed two-fluid SPH scheme in case of stiff linear drag thanks to the fact that spatial resolution condition h < c s t stop is not required anymore for accurate results.

Fast method to simulate dynamics of two-phase medium with intense interaction between phases by smoothed particle hydrodynamics: Gas-dust mixture with polydisperse particles, linear drag, one-dimensional tests

“…As a result, in the calculation of intense drag, a numerical overdissipation appears. This overdissipation was demonstrated and discussed in papers [38,34,35]. Laibe and Price [38] showed that the scale of this overdissipation depends on stopping time.…”

“…Obviously, the computational costs and the asymptotic properties of the method will depend on the way the relative velocities in the relaxation terms are calculated. To date, three ways of spatial velocity interpolation have been developed for the smoothed particle hydrodynamics, which we will call particle-particle [33], duplicate node [34] and drag in a cell [35].…”

Simulations of Dynamical Gas–Dust Circumstellar Disks: Going Beyond the Epstein Regime

“…Stoyanovskaya et al [38] proposed a new method for solving the motion equations of gas and monodisperse dust with intense interphase interactions based on SPH. Their proposed method for computing the interphase interactions is non-iterative and has low dissipative properties.…”

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

“…In addition, computing of the drag force can be based on the idea of the Particle-in-Cell method for simulation of gas-dust flows [21]. The fast semi-implicit SPH-IDIC approach based on this idea was suggested and tested in our earlier paper [8]. A detailed description of this approach is presented below.…”

“…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.…”