Abstract
For operator splitting methods, an approach based on Taylor expansion and the particular structure of its leading term as an element of a free Lie algebra is used for the setup of a system of order conditions. Along with a brief review of the underlying theoretical background, we discuss the implementation of the resulting algorithm in computer algebra, in particular using Maple 18 (Maple is a trademark of MapleSoft\(^\mathrm{TM}\).). A parallel version of such a code is described, and its performance on a computational node with 16 threads is documented.
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- 1.
The aspect (ii) enters the discussion in [2].
- 2.
By \(\varPhi _F\) we denote the flow associated with the equation \(\partial _t u = F(u)\), and \(\varPhi _A,\,\varPhi _B\) are defined analogously.
- 3.
‘Non-vanishing’ means non-vanishing in general (generic case, with no special assumptions on A and B).
- 4.
If A and B commute, i.e., \(AB=BA\), then all these expressions vanish.
- 5.
There are special cases of practical interest where this growth is much more moderate; we do not discuss such details here.
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Acknowledgements
This work was supported by the Austrian Science Fund (FWF) under grant P24157-N13, and by the Vienna Science and Technology Fund (WWTF) under grant MA-14-002. Computational results based on the ideas in this work have been achieved in part using the Vienna Scientific Cluster (VSC).
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Auzinger, W., Herfort, W., Hofstätter, H., Koch, O. (2016). Setup of Order Conditions for Splitting Methods. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_3
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DOI: https://doi.org/10.1007/978-3-319-45641-6_3
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