scholarly journals Embedded operator splitting methods for perturbed systems

2020 ◽  
Vol 492 (4) ◽  
pp. 5413-5419
Author(s):  
Hanno Rein

ABSTRACT It is common in classical mechanics to encounter systems whose Hamiltonian H is the sum of an often exactly integrable Hamiltonian H0 and a small perturbation ϵH1 with ϵ ≪ 1. Such near-integrability can be exploited to construct particularly accurate operator splitting methods to solve the equations of motion of H. However, in many cases, for example in problems related to planetary motion, it is computationally expensive to obtain the exact solution to H0. In this paper, we present a new family of embedded operator splitting (EOS) methods which do not use the exact solution to H0, but rather approximate it with yet another, EOS method. Our new methods have all the desirable properties of classical methods which solve H0 directly. But in addition they are very easy to implement and in some cases faster. When applied to the problem of planetary motion, our EOS methods have error scalings identical to that of the often used Wisdom–Holman method but do not require a Kepler solver, nor any coordinate transformations, or the allocation of memory. The only two problem specific functions that need to be implemented are the straightforward kick and drift steps typically used in the standard second-order leap-frog method.

2019 ◽  
Vol 180 (1-2) ◽  
pp. 489-532 ◽  
Author(s):  
Yang Zheng ◽  
Giovanni Fantuzzi ◽  
Antonis Papachristodoulou ◽  
Paul Goulart ◽  
Andrew Wynn

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jürgen Geiser

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.


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