Stability and convergence of numerical computations

dissertation

English

The aim of this thesis is to analyze the stability and convergence of fundamental numerical methods for solving ordinary differential equations. 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 main goal and the new contribution of the thesis 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. The mathematical background of historical developments of Nordsieck representation, the algorithm of choosing a variable stepsize or an error estimation are discussed. The current approach adapts well to the multiderivative situation in variable stepsize formulations. Experiments for linear and non-linear problems and the comparison with classical methods are presented.

Numerical methods, ordinary differential equations, stability, convergence, one-step methods, Runge-Kutta methods, Taylor series method, multistep methods, linear multistep methods, Adams methods, predictor-corrector pairs, Obreshkov formulae, Nordsieck representation, variable stepsize.

SEHNALOVÁ, P.

27. 9. 2011

Brno

108

https://www.fit.vut.cz/research/publication/9778/

PHDthesis_Sehnalova.pdf