Volume preservation by Runge–Kutta methods
McLaren, David I.
Quispel, G. R. W.
Applied Numerical Mathematics
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Bader, P., McLaren, D. I., Quispel, G. R. W., & Webb, M. (2016). Volume preservation by Runge–Kutta methods. Applied Numerical Mathematics, 109 123-137. https://doi.org/10.1016/j.apnum.2016.06.010
This is the final version of the article. It first appeared from Elsevier via http://dx.doi.org/10.1016/j.apnum.2016.06.010
It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge–Kutta method will respect this property for such systems, but it has been shown by Iserles, Quispel and Tse and independently by Chartier and Murua that no B-Series method can be volume preserving for all volume preserving vector fields. In this paper, we show that despite this result, symplectic Runge–Kutta methods can be volume preserving for a much larger class of vector fields than Hamiltonian systems, and discuss how some Runge–Kutta methods can preserve a modified measure exactly.
volume preservation, Runge–Kutta method, measure preservation, Kahan’s method
This research was supported by the Marie Curie International Research Staff Exchange Scheme, grant number DP140100640, within the 7th European Community Framework Programme; by the Australian Research Council grant number 269281; and by the UK Engineering and Physical Sciences Research Council grant EP/H023348/1 for the Cambridge Centre for Analysis.
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External DOI: https://doi.org/10.1016/j.apnum.2016.06.010
This record's URL: https://www.repository.cam.ac.uk/handle/1810/257255
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Licence URL: http://creativecommons.org/licenses/by/4.0/
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