Two-step Runge-Kutta Methods for Ordinary and Stochastic Differential Equations

The aim of this thesis is the derivation and the analysis of high order efficient stable methods, both for Ordinary Differential Equations (ODEs) and Stochastic Differential Equations (SDEs), which are models of many important phenomena in life science. In chapter 1 we recall, just for completeness, some preliminary notions on Runge-Kutta methods, Two-step Runge-Kutta methods and General Linear Methods. In the second chapter we introduce a new class of two-step collocation methods for first order ODEs. We derive continuous order conditions, proving that our new methods have uniform order 2m+1, where m is the number of stages. We carry out the linear stability analysis, but we did not found A-stable methods within this class. In order to improve the stability properties, we relax the collocation technique and we obtain A-stable and L-stable methods. At the end we give some numerical experiments, in order to confirm the theoretical properties of the new classes of methods. In the third chapter we derive collocation hybrid methods, with constant coefficients, for special second order ordinary differential equations having periodic or oscillatory solutions. As we done for first order ODEs, we derive continuous order conditions and analyse the linear stability properties. Then we adapt the coefficients of the two-step hybrid method to an oscillatory behaviour, in such a way that it exactly integrates linear combinations of power and trigonometric functions depending on one and two frequencies, which we suppose can be estimated in advance. Frequency-dependent methods within this class have already been considered in, where the coefficients of methods were modified to produce phase-fitted and amplification-fitted methods. We show the constructive technique of methods based on trigonometric and mixed polynomial fitting and consider the linear stability analysis of such methods. Then we carry out some numerical experiments underlining the properties of the derived classes of methods. The fourth chapter is dedicated to an introduction to stochastic ordinary differential equations. Since this is quite a new topic, we will recall the It^o calculus and the properties of multiple stochastic integrals. The chapter 5 is concerned to the derivation of order conditions for a new class of Stochastic Runge--Kutta methods, by an extension of Albrecht approach for SDEs. We rewrite our nonlinear Runge--Kutta method as a composition of linear multistep methods. We proceed as done in the linear case in order to derive order conditions also for the internal stages. We also show the advantages in the convergence framework.