Abstract
We present new, explicit, volume-preserving splitting methods for polynomial divergence-free vector fields of arbitrary degree (both positive and negative). The main idea is to decompose the divergence polynomial by means of an appropriate basis for polynomials: the monomial basis. For each monomial basis function, the split fields are then identified by collecting the appropriate terms in the vector field so that each split vector field is volume preserving. We show that each split field can be integrated exactly by analytical methods. Thus, the composition yields a volume preserving numerical method. Our numerical tests indicate that the methods compare favorably to standard integrators both in the quality of the numerical solution and the computational effort.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blanes, S., Moan, P.C.: Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. J. Comput. Appl. Math. 142, 313–330 (2002)
Chartier, P., Murua, A.: Preserving first integrals and volume forms of additively split systems. IMA J. Numer. Anal. 27(2), 381–405 (2007)
Dragt, A.J., Abell, D.T.: Symplectic maps and computation of orbits in particle accelerators. In: Integration Algorithms and Classical Mechanics, Toronto, ON, 1993. Fields Inst. Commun., vol. 10, pp. 59–85. Am. Math. Soc., Providence (1996)
Iserles, A., Quispel, G.R.W., Tse, P.S.P.: B-series methods cannot be volume-preserving. BIT Numer. Math. 47(2), 351–378 (2007)
Kang, F., Shang, Z.J.: Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 71(4), 451–463 (1995)
Mclachlan, R.I., Scovel, C.: A survey of open problems in symplectic integration. Fields Inst. Commun. 10, 151–180 (1998)
McLachlan, R.I., Quispel, G.R.W.: Explicit geometric integration of polynomial vector fields. BIT Numer. Math. 44, 515–538 (2004)
McLachlan, R.I., Munthe-Kaas, H.Z., Quispel, G.R.W., Zanna, A.: Explicit volume-preserving splitting methods for linear and quadratic divergence-free vector fields. Found. Comput. Math. 8(3), 335–355 (2008)
Neishtadt, A., Vainchtein, D., Vasiliev, A.: Adiabatic invariance in volume-preserving systems. In: IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence. IUTAM Bookser., vol. 6, pp. 89–107. Springer, Dordrecht (2008)
Qin, M.-Z., Zhu, W.-J.: Volume-preserving schemes and numerical experiments. Comput. Math. Appl. 26, 33–42 (1993)
Quispel, G.R.W.: Volume-preserving integrators. Phys. Lett. A 206(1–2), 26–30 (1995)
Rangarajan, G.: Symplectic completion of symplectic jets. J. Math. Phys. 37(9), 4514–4542 (1996)
Scovel, J.C.: Symplectic numerical integration of Hamiltonian systems. In: Ratiu, T. (ed.) The Geometry of Hamiltonian Systems. MSRI, vol. 22, pp. 463–496. Springer, New York (1991)
Shang, Z.J.: Construction of volume-preserving difference schemes for source-free systems via generating functions. J. Comput. Math. 12(3), 265–272 (1994)
Stone, H.A., Nadim, A., Strogatz, S.H.: Chaotic streamlines inside drops immersed in steady Stokes flows. J. Fluid Mech. 232, 629–646 (1991)
Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990)
Acknowledgements
This work was supported by NFR grant GeNuIn: Geometric Numeric Integration in Applications, project no. 191178/V30.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mechthid Thalhammer.
Rights and permissions
About this article
Cite this article
Xue, H., Zanna, A. Explicit volume-preserving splitting methods for polynomial divergence-free vector fields. Bit Numer Math 53, 265–281 (2013). https://doi.org/10.1007/s10543-012-0394-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-012-0394-0