On a Hermite Integrator with Ahmad-Cohen Scheme for Gravitational Many-Body Problems

We describe the implementation of the Ahmad-Cohen scheme based on a fourth-order Hermite integrator. With the fourth-order Hermite scheme, we calculate the force and the time derivative of the force analytically, and construct a third-order interpolation polynomial using two points in time. Compared with the standard scheme (Aarseth 1985) which is widely used, it allows a longer stepsize for the same accuracy, and the program is much simpler. In the case of the Ahmad-Cohen scheme, which uses different stepsides for the forces from neighboring particles and that from distant particles, the difference in the programming complexity is even larger, since the Hermite scheme does not require corrections of the higher order divided diferences for the forces from distant particles. On scalar computes the Hermite schemes are marginally faster than the standard scheme for the same level of accuracy, both with and without the Ahmad-Cohen scheme. On vector machines or special-purpose hardware, such as GRAPE, the Hermite scheme would be significantly faster since the number of scalar operation is much smaller. The gain in computing speed using the Ahmad-Cohen scheme is (N/3.8)(1/4) for both the standard and Hermite schemes, where N is the total number of particles. However, this gain can be significantly smaller on vector or parallel machines.