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International Journal of Differential Equations
Volume 2013 (2013), Article ID 314290, 11 pages
http://dx.doi.org/10.1155/2013/314290
Research Article

Embedded Zassenhaus Expansion to Splitting Schemes: Theory and Multiphysics Applications

Institute of Physics, Felix-Hausdorff-Street 6, 17489 Greifswald, Germany

Received 25 March 2013; Revised 5 August 2013; Accepted 19 August 2013

Academic Editor: Shuyu Sun

Copyright © 2013 Jürgen Geiser. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present some operator splitting methods improved by the use of the Zassenhaus product and designed for applications to multiphysics problems. We treat iterative splitting methods that can be improved by means of the Zassenhaus product formula, which is a sequential splitting scheme. The main idea for reducing the computation time needed by the iterative scheme is to embed fast and cheap Zassenhaus product schemes, since the computation of the commutators involved is very cheap, since we are dealing with nilpotent matrices. We discuss the coupling ideas of iterative and sequential splitting techniques and their convergence. While the iterative splitting schemes converge slowly in their first iterative steps, we improve the initial convergence rates by embedding the Zassenhaus product formula. The applications are to multiphysics problems in fluid dynamics. We consider phase models in computational fluid dynamics and analyse how to obtain higher order operator splitting methods based on the Zassenhaus product. The computational benefits derive from the use of sparse matrices, which arise from the spatial discretisation of the underlying partial differential equations. Since the Zassenhaus formula requires nearly constant CPU time due to its sparse commutators, we have accelerated the iterative splitting schemes.