The following project deals with two main topics: General Linear methods (GLM) and jet transport. For their presentation, we have divided it in three chapters. In Chapter 1, we introduce the family of numerical integrators known as General Linear methods, which arise as a natural generalization of the so-known linear multistep (LMM) and Runge-Kutta (RK) methods. Throughout the chapter, we present the main properties of LMM and RK methods so that they can be compared with those obtained for GLM with greater generality. In Chapter 2, we introduce the technique known as jet transport for the numerical integration of variational equations. It is in this chapter where the main contributuion of this project is found: we prove that the numerical integration of an initial value problem using jet transport with General Linear methods is equivalent to the numerical integration of their variational equations with the same method. Not only that, but we also successfully derive the expressions that the higher order coefficients of the jets must satisfy to be a solution of an implicit system, thus allowing the effective implementation of implicit General Linear methods. In Chapter 3 we conclude this project by studying how implicit Runge-Kutta methods can be efficiently implemented using jet transport and we apply this implementation to study a few scenarios in the field of dynamical systems, where the computation of variational equations is of interest.

Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Curs: 2022-2023. Director: Àngel Jorba i Monte; Joan Gimeno Alquezar