Shear flows in smoothed particle hydrodynamics

Evolutionary calculations of rotating gaseous flows around astrophysical objects with standard smoothed particle hydrodynamics (SPH) result in inaccurate evolutions of shear flows. The large density errors (Deltarhogreater than or similar torho) emerge within one dynamical time of the system when we adopt the mean separation of the particles as a smoothing length, h. We show that these errors originate in the definition of density and the inaccurate solution of the continuity equation in standard SPH. Standard SPH ensures the global mass conservation but does not ensure the local mass conservation. Thus, we reformulate the SPH method and find the expression of the consistent velocity field in terms of particle velocities that ensure local mass conservation. We update the particle positions using particle velocities instead of fluid velocities. These two velocities should be different theoretically. The numerical examples of the improved method are shown. The implementation of SPH described in this paper is crucial for the long-term calculations of the shear flows in differentially rotating stars, circumstellar disks, and galactic gaseous disks.