Local error analysis for generalised splitting methods

Abstract

This thesis introduces the basic theory of splitting methods for evolution equations. A symmetrised version of the defect is discussed and the defect is established as an asymptotically correct local error estimator. The general background for high order splitting such as the Baker-Campbell-Hausdorff formula and symmetrised versions there of are treated and order conditions for high order splittings are extracted. In particular, we take a closer look at skew-hermitian matrices. In addition, we cover a 'dual' approach - the Zassenhaus splitting - and discuss the main ingredients Magnus provided for the analysis of the Zassenhaus splitting. A symmetrisation of Magnus' approach is made. Next, we introduce inner symmetrised defects and elaborate on its Taylor expansion. This is the key component to the more basic approach. We focus on the error expansion of the Strang splitting - our basic case of the more general Zassenhaus type setting. The systematic treatment of the general case offers ideas for further generalisations and provides a basis for a good understanding of the high level theory results. It is based on the Faà di Bruno identity and Bell polynomials play a key role when generalising the Lie expansion formula. We use Feynman diagrams for a compact and clear picture of the derivatives we will encounter. In the end, we have successfully recovered the order condition previously seen in the BCH formula by using the Taylor approach. We will conclude the thesis with an application of the order conditions to a physical problem.