  We study and compare different numerical differential equation solvers on the
basis of numerical complexity, energy conservation, and stable solution in
phase-space for the Simple Harmonic Oscillation (SHM) problem. We conclude and
show that the Leapfrog method is the best for our problem. We solve Poisson's
equation in gravity on the computation grid by the finite difference method and
Fourier method. We solve Poisson's equation for source not in a grid point by
cloud-in cell method. Finally, we simulate one-dimensional self-gravitating
system and show the evolution of the system via phase-space trajectories.
