PARTICLE–GAS DYNAMICS WITH ATHENA: METHOD AND CONVERGENCE

The Athena magnetohydrodynamics code has been extended to integrate the motion of particles coupled with the gas via aerodynamic drag in order to study the dynamics of gas and solids in protoplanetary disks (PPDs) and the formation of planetesimals. Our particle–gas hybrid scheme is based on a second-order predictor–corrector method. Careful treatment of the momentum feedback on the gas guarantees exact conservation. The hybrid scheme is stable and convergent in most regimes relevant to PPDs. We describe a semi-implicit integrator generalized from the leap-frog approach. In the absence of drag force, it preserves the geometric properties of a particle orbit. We also present a fully implicit integrator that is unconditionally stable for all regimes of particle–gas coupling. Using our hybrid code, we study the numerical convergence of the nonlinear saturated state of the streaming instability. We find that gas flow properties are well converged with modest grid resolution (128 cells per pressure length ηr for dimensionless stopping time τ s = 0.1) and an equal number of particles and grid cells. On the other hand, particle clumping properties converge only at higher resolutions, and finer resolution leads to stronger clumping before convergence is reached. Finally, we find that the measurement of particle transport properties resulted from the streaming instability may be subject to error of about ±20%.