Dusty gas with one fluid in smoothed particle hydrodynamics

In a companion paper we have shown how the equations describing gas and dust as two fluids coupled by a drag term can be re-formulated to describe the system as a single-fluid mixture. Here, we present a numerical implementation of the one-fluid dusty gas algorithm using smoothed particle hydrodynamics (SPH). The algorithm preserves the conservation properties of the SPH formalism. In particular, the total gas and dust mass, momentum, angular momentum and energy are all exactly conserved. Shock viscosity and conductivity terms are generalized to handle the two-phase mixture accordingly. The algorithm is benchmarked against a comprehensive suit of problems: dustybox, dustywave, dustyshock and dustyoscill, each of them addressing different properties of the method. We compare the performance of the one-fluid algorithm to the standard two-fluid approach. The one-fluid algorithm is found to solve both of the fundamental limitations of the two-fluid algorithm: it is no longer possible to concentrate dust below the resolution of the gas (they have the same resolution by definition), and the spatial resolution criterion h < c(s)t(s), required in two-fluid codes to avoid over-damping of kinetic energy, is unnecessary. Implicit time-stepping is straightforward. As a result, the algorithm is up to ten billion times more efficient for 3D simulations of small grains. Additional benefits include the use of half as many particles, a single kernel and fewer SPH interpolations. The only limitation is that it does not capture multi-streaming of dust in the limit of zero coupling, suggesting that in this case a hybrid approach may be required.