In this thesis, numerical methods for a certain
class of singular initial value problems of the second order
are investigated. The second order problems are transformed to first
order systems with a singularity of the first kind and solved
using one-step methods (implicit and explicit Euler method,
trapezoidal rule and box scheme). It is proven that the implicit and
explicit Euler method and the trapezoidal rule show their classical
convergence properties (order 1 and 2, respectively), whereas
for the box scheme an order reduction occurs which can also be observed
in experiments. Finally, an asymptotic
error expansion for the implicit Euler method is derived, which is
frequently a basis for the use of certain acceleration techniques.
The solvability of a certain class of singular nonlinear initial value
problems is discussed. Particular attention is paid to the structure of
initial conditions necessary for a bounded solution to exist. The implicit
Euler rule applied to approximate the solution of the singular system is
shown to be stable and to retain its classical convergence
order. Moreover, the asymptotic error expansion for the global error of the
above approximation is proven to have the classical structure. Finally,
experimental results showing the feasibility of the approximation obtained
by the Euler method to serve as a basic method for the acceleration
technique known as the Iterated Defect Correction are presented.