The Athena MHD code has been extended to integrates the motion of particles coupled with the gas via aerodynamic drag , in order to study the dynamics of gas and solids in protoplanetary disks 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 protoplanetary disks .
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 non-linear saturated state of the streaming instability .
We find that gas flow properties are well converged with modest grid resolution ( 128 cells per pressure length \eta r for dimensionless stopping time \tau _ { s } = 0.1 ) , and 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 measurement of particle transport properties resulted from the streaming instability may be subject to error of about \pm 20 \% .