Dusty gas with smoothed particle hydrodynamics - II. Implicit timestepping and astrophysical drag regimes

In a companion paper, we have presented an algorithm for simulating two-fluid gas and dust mixtures in smoothed particle hydrodynamics (SPH). In this paper, we develop an implicit timestepping method that preserves the exact conservation of the both linear and angular momenta in the underlying SPH algorithm, but unlike previous schemes, allows the iterations to converge to arbitrary accuracy and is suited to the treatment of non-linear drag regimes. The algorithm presented in Paper I is also extended to deal with realistic astrophysical drag regimes, including both linear and non-linear Epstein and Stokes drag. The scheme is benchmarked against the test suite presented in Paper I, including (i) the analytic solutions of the dustybox problem and (ii) solutions of the dustywave, dustyshock, dustysedov and dustydisc obtained with explicit timestepping. We find that the implicit method is 110 times faster than the explicit temporal integration when the ratio r between the timestep and the drag stopping time is 1 ?r? 1000.