An analysis of block Verlet timestepping and lagged force evaluations for the gravitational N-body problem

N-body simulations are widely used in astrophysics to model the behavior of stellar system. The Verlet integrator is one of the principle numerical methods used to solve the N-body problem. It is famous because of its significant features such as good numerical stability, time reversibility, preservation of the symplectic form on phase space and economy of memory. Since it is symplectic, it exactly solves an approximate Hamiltonian. The Verlet integrator preserves certain conserved quantities exactly, such as the total angular and linear momentum and the phase-space volume.

Since the density in a stellar system varies with distance from its center, it is extremely inefficient to integrate all the stars with the same timestep, so an integrator should allow individual timestep for each star. The block timestep scheme will help to do this.

We have analyzed both Verlet and block timestep Verlet method. We proved that block timestep Verlet method has GE, O(h^2). The block timestep Verlet method has same advantages as in Verlet method except the symplecticity. We have seen that the block timestep Verlet method more efficient than regular Verlet method. We carry out experiments to check our analysis.

When developing N- body simulation, the force calculation is one of the important factors. The best numerical integrator for a problem is evaluate the force cheaply. We introduced a method called lagged calculation using the functional expansion method.