Aspectos Teóricos dos Métodos Runge-Kutta

Abstract

Neste traballo presentaremos diferentes propiedades e resultados referidos aos métodos Runge-Kutta. Comezaremos co estudo dos métodos dun paso, abordando a consistencia, estabilidade, converxencia e orde de devanditos métodos. Tras isto, centrarémonos no estudo destes conceptos nunha familia concreta dos métodos dun paso, que son os chamados métodos Runge-Kutta. Finalmente, estudaremos algúns aspectos teóricos relativos á estabilidade numérica dos métodos Runge-Kutta, que introduciremos en primeiro lugar no caso dos métodos de Euler implícito e explícito, para logo abordar estes aspectos no caso dun método Runge-Kutta xeral.

In this project we will present different properties and results concerning Runge-Kutta methods. We wil start with the study of one-step methods, addressing the consistency, stability, convergence and order of such methods. After this, we will focus on the study of these concepts in a particular family of one-step methods, which are the so-called Runge-Kutta methods. Finally, we will study some theoretical aspects concerning the numerical stability of Runge-Kutta methods, which we will first introduce in the case of implicit and explicit Euler methods, and then address aspects in the case of a general Runge-Kutta method.