The term differential-algebraic equations (DAE's) comprises all systems of equations of the form R(y', y, x) = 0 where [the derivative of R with respect to y'] is not of full rank. The formulation of numerous problems and the modelling of many physical systems lead in a natural way to such equations.

This thesis deals more specifically with a class of DAE's, called semi-explicit index 3 DAE's in Hessenberg form, whose various application domains are given by mechanical and Hamiltonian systems with constraints, molecular dynamics, optimal control problems, and robotics. The properties of numerical methods of Runge-Kutta type are studied. We largely make use of tree-structures which appear in the Taylor-expansion of the exact and numerical solutions.

A main result of this thesis is a demonstration of a conjecture related to the superconvergence of stiffiy accurate Runge-Kutta methods. Also interesting is the discovery of a class of partitioned Runge-Kutta methods which, when applied to Hamiltonian systems with constraints, allow to preserve the constraints and the symplectic structure of the flow.