Numerische Integration nichtlinearer Evolutionsgleichungen : Entwicklung und Implementierung verschiedener numerischer Verfahren

Abstract

This diploma thesis deals with the analysis and the numerical implementation of integration schemes for evolution equations, as well as asymptotically correct defect-based local error estimators. Recent work in [Auzinger, Koch, Hofstätter (2018)] introduced the symmetrized defect for the linear, autonomous Schrödinger-type evolution equation, which proved to be of a higher asymptotical order than the well known classical defect, in the case of symmetric time-stepping. In [Auzinger, Koch (2018)] this result was proven to hold true also for nonlinear and nonautonomous evolution equations. In this thesis we recapitulate these results in a concise, but complete manner, whereby we set citations in a more general then detailed way. After that, we apply them to established, as well as newly constructed splitting schemes. To provide numerical verification of the theory, we consider several linear/nonlinear, autonomous/nonautonomous model problems.