Embedded Zassenhaus Expansion to Splitting Schemes: Theory and Multiphysics Applications

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. 1. Introduction Our motivation to study the operator splitting methods comes from models in fluid dynamics, for example problems in bioremediation [1] or radioactive contaminants [2]. Such multiphysics problems are delicate, and the solver methods can be accelerated by decoupling the different physical behaviours; see [3, 4]. Based on the splitting error of all the splitting schemes, we have to take into account higher order ideas; see [5, 6]. While standard splitting methods deal with lower order convergence, see Lie splitting and Strang splitting [7, 8], we propose a combination of iterative splitting methods with an embedded Zassenhaus product formula. Such a combination allows reducing the splitting error and also reducing the CPU time needed; see [9]. Theoretically, we combine fixed point schemes (iterative splitting methods) with sequential splitting schemes (Zassenhaus products), which are connected with the theory of Lie groups and Lie algebras. Based on that relation, we can construct higher order splitting schemes for an underlying Lie algebra and improve the convergence results with cheap iterative schemes. Historically, the efficiency of decoupling different physical processes into simpler processes, for example, convection and