  We describe a new algorithm for the integration of self-gravitating fluid
systems using SPH method. We split the Hamiltonian of a self-gravitating fluid
system to the gravitational potential and others (kinetic and internal
energies) and use different time-steps for their integrations. The time
integration is done in the way similar to that used in the mixed variable or
multiple stepsize symplectic schemes. We performed three test calculations. One
was the spherical collapse and the other was an explosion. We also performed a
realistic test, in which the initial model was taken from a simulation of
merging galaxies. In all test calculations, we found that the number of
time-steps for gravitational interaction were reduced by nearly an order of
magnitude when we adopted our integration method. In the case of the realistic
test, in which the dark matter potential dominates the total system, the total
calculation time was significantly reduced. Simulation results were almost the
same with those of simulations with the ordinary individual time-step method.
Our new method achieves good performance without sacrificing the accuracy of
the time integration.
