The hermite scheme: an application to the n-body problem

Abstract

In the past century, computational methods have been being applied more and more to physical systems, in special to systems which are chaotic in nature or don’t have an analytical solution, or both, such as is the case for systems that obey the N-body problem. To solve such systems, it is necessary to select the most suitable numerical method, one that takes into account both the necessary time and computational resources available to the researcher, and in order to be able to do so, one must have a good set of tools available. In this work we present a numerical method known as the Hermite Scheme, a fourth-order predictor-corrector integrator which makes use of an Individual Time Step structure, making it capable of processing multi-scale systems. We test its accuracy and study its applicability to the N-body problem, extending the result to chaotic systems in general. We then proceed to check its performance for a N-body system, and compare it to the performance of another fourth-order integrator, the Runge-Kutta. Lastly we verify its performance to multi-scale systems by reproducing some real-life results. Our results show that the Hermite Scheme has a good applicability to N-body systems, with an overall performance better than the fourth order Runge-Kutta. It also shows a good performance when applied to multi-scale systems, with no harm to its overall time performance, which can be applied to other multi-scale systems other than the N-body problem. With this verification, we intend to further apply this method to collision processes and apply the final result on the study of planet formation. The method also shows great potential applicability to Condensed Matter Physics, and we intend to test-apply to known systems in the future.