Efficient Numerical Integration for Gravitational N-Body Simulations

Abstract

Models for N-body gravitational simulations of the Solar System vary from small simulations of two bodies over short intervals of time to simulations of large numbers of bodies over long-term integration. Most simulations require the numerical solution of an initial value problem (IVP) of second-order ordinary differential equation. We present new integration methods intended for accurate simulations that are more efficient than existing methods. In the first part of the thesis, we present new higher-order explicit Runge–Kutta Nyström pairs. These new pairs are searched using a simulated annealing algorithm based on optimisation. The new pairs are up to approximately 60% more efficient than the existing ones. We implement these new pairs for a variety of gravitational problems and investigate the growth of global error in position for these problems along with relative error in conserved quantities. The second part consists of the implementation of the Gauss Implicit Runge-Kutta methods in an efficient way such that the error growth satisfies Brouwer’s Law. Numerical experiments show that using the new way of implementation reduces the integration cost up to 20%. We also implement continuous extensions for the Gauss implicit Runge-Kutta methods, using interpolation polynomials at nodal points.