  We present a novel method for particle splitting in smoothed particle
hydrodynamics simulations. Our method utilizes the Voronoi diagram for a given
particle set to determine the position of fine daughter particles. We perform
several test simulations to compare our method with a conventional splitting
method in which the daughter particles are placed isotropically over the local
smoothing length. We show that, with our method, the density deviation after
splitting is reduced by a factor of about two compared with the conventional
method. Splitting would smooth out the anisotropic density structure if the
daughters are distributed isotropically, but our scheme allows the daughter
particles to trace the original density distribution with length scales of the
mean separation of their parent. We apply the particle splitting to simulations
of the primordial gas cloud collapse. The thermal evolution is accurately
followed to the hydrogen number density of 10^12 /cc. With the effective mass
resolution of ~10^-4 Msun after the multi-step particle splitting, the
protostellar disk structure is well resolved. We conclude that the method
offers an efficient way to simulate the evolution of an interstellar gas and
the formation of stars.
