Stability and Convergence of Numerical Computations

The stability and convergence of fundamental numerical methods for solving ordinary differential equations are presented. These include one-step methods such as the classical Euler method, Runge–Kutta methods and the less well known but fast and accurate Taylor series method. We also consider the generalization to multistep methods such as Adams methods and their implementation as predictor–corrector pairs. Furthermore we consider the generalization to multiderivative methods such as Obreshkov method. There is always a choice in predictor-corrector pairs of the so-called mode of the method and in this thesis both PEC and PECE modes are considered. The aim of the paper is the use of a special fourth order method consisting of a two-step predictor followed by an one-step corrector, each using second derivative formulae and the convergence and stability analysis for the new method with constant stepsize for various problems as well as to investigate and to compare the convergence and stability analysis for selected numerical methods. Experiments for linear and non-linear problems and the comparison with classical methods are presented.