Mètodes numèrics per equacions diferencials aplicats a mecànica celeste

Abstract

[en] In this document, we will explore a selection of interesting numerical methods for ordinary differential equations. Mainly, we will explain the Taylor methods, the Runge-Kutta methods and the Extrapolation methods, providing respectively a theoretical basis, where we will delve into key aspects of numerical methods, including convergence or stability. In general, our aim will be to guide the theory to the implementation of the methods in programs in C language using advanced control techniques on step, that will allow us to obtain results with errors below a predetermined tolerance. For this reason, our primary focus will be on the concrete methods of Taylor applying automatic differentiation, Runge-Kutta-Fehlberg methods and the Extrapolation Gragg-Burlisch-Stoer method. To demonstrate the practicality of these approaches, we will apply them to some celestial mechanics problems, such as the Central Force Problem or a more general version, the N-Body Problem. By utilizing actual data from the Solar System, we will compare the accuracy and efficiency of these three methods drawing appropriate conclusions.