Direct Integration of the Collisionless Boltzmann Equation in Six-dimensional Phase Space: Self-gravitating Systems

We present a scheme for numerical simulations of collisionless self-gravitating systems which directly integrates the Vlasov--Poisson equations in six-dimensional phase space. By the results from a suite of large-scale numerical simulations, we demonstrate that the present scheme can simulate collisionless self-gravitating systems properly. The integration scheme is based on the positive flux conservation method recently developed in plasma physics. We test the accuracy of our code by performing several test calculations including the stability of King spheres, the gravitational instability and the Landau damping. We show that the mass and the energy are accurately conserved for all the test cases we study. The results are in good agreement with linear theory predictions and/or analytic solutions. The distribution function keeps the property of positivity and remains non-oscillatory. The largest simulations are run on 64^6 grids. The computation speed scales well with the number of processors, and thus our code performs efficiently on massively parallel supercomputers.