On the preservation of second integrals by Runge-Kutta methods

Benjamin K. Tapley

doi:10.3934/jcd.2023001

Article Contents

2023, Volume 10, Issue 2: 304-322. Doi: 10.3934/jcd.2023001

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On the preservation of second integrals by Runge-Kutta methods

Benjamin K. Tapley^1,2,

1.
Department of Mathematics and Cybernetics, SINTEF Digital, 0373 Oslo, Norway
2.
Department of Mathematical Sciences, The Norwegian University of Science and Technology, 7491 Trondheim, Norway

Received: July 2021

Revised: January 2023

Early access: March 2023

Published: April 2023

This work has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement (No. 691070)

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Abstract

One can elucidate integrability properties of ordinary differential equations (ODEs) by knowing the existence of second integrals (also known as weak integrals or Darboux polynomials for polynomial ODEs). However, little is known about how they are preserved, if at all, under numerical methods. Here, we leverage the recently discovered theory of discrete second integrals to show novel results about Runge-Kutta methods. In particular, we show that any Runge-Kutta method preserves all affine second integrals but cannot preserve all quadratic second integrals of an ODE. A number of interesting corollaries are also discussed, such as the preservation of certain rational integrals by Runge-Kutta methods. The special case of affine second integrals with constant cofactor are also discussed as well the preservation of third and higher integrals.

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Mathematics Subject Classification: Primary: 37M05, Secondary: 65L06, 65P10.

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Figure 1. The phase portrait of the ODE (15) and the errors of the second integrals $ p_i( \boldsymbol{x}) $ for initial conditions satisfying $ p_i( \boldsymbol{x}_0) = 0 $. Note that $ p_3( \boldsymbol{x}_n)-p_3( \boldsymbol{x}_0) = 0 $ and therefore does not show on the semi-log axis. The initial conditions are shown by black dots and are located on the grid $ (-10 + i, -10+j) $ for $ i, j = 0, ..., 20 $

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References

[1]	D. W. Albrecht, E. L. Mansfield and A. E. Milne, Algorithms for special integrals of ordinary differential equations, J. Phys. A, 29 (1996), 973-991. doi: 10.1088/0305-4470/29/5/013.
[2]	E. Celledoni, C. Evripidou, D. I. McLaren, B. Owren, G. R. W. Quispel and B. K. Tapley, Detecting and determining preserved measures and integrals of birational maps, J. Comput. Dyn., 9 (2022), 553-574. doi: 10.3934/jcd.2022014.
[3]	E. Celledoni, C. Evripidou, D. I. McLaren, B. Owren, G. R. W. Quispel, B. K. Tapley and P. H. van der Kamp, Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps, J. Phys. A, 52 (2019), 31LT01, 11 pp. doi: 10.1088/1751-8121/ab294b.
[4]	E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren, G. Reinout W Quispel and W. M. Wright, Energy-preserving Runge-Kutta methods, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), 645-649. doi: 10.1051/m2an/2009020.
[5]	E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel, Geometric properties of Kahan's method, J. Phys. A, 46 (2012), 025201. doi: 10.1088/1751-8113/46/2/025201.
[6]	G. J. Cooper, Stability of Runge-Kutta methods for trajectory problems, IMA J. Numer. Anal., 7 (1987), 1-13. doi: 10.1093/imanum/7.1.1.
[7]	C. B. Collins, Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree, J. Math. Anal. Appl., 195 (1995), 719-735. doi: 10.1006/jmaa.1995.1385.
[8]	M. Dahlby, B. Owren and T. Yaguchi, Preserving multiple first integrals by discrete gradients, J. Phys. A: Mathematical and Theoretical, 44 (2011), 305205, 14 pp. doi: 10.1088/1751-8113/44/30/305205.
[9]	A. Goriely, Integrability and Nonintegrability of Dynamical Systems, volume 19., World Scientific, 2001. doi: 10.1142/9789812811943.
[10]	E. Hairer, Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5 (2010), 73-84.
[11]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, volume 31., 2 edition, Springer-Verlag, Berlin, 2006.
[12]	C. G. Hewitt, Algebraic invariant curves in cosmological dynamical systems and exact solutions, Gen. Relativity Gravitation, 23 (1991), 1363-1383. doi: 10.1007/BF00756830.
[13]	J.-P. Jouanolou, Equations de Pfaff Algébriques, volume 708, Springer, Berlin, 1979.
[14]	F. Kang and S. Zai-jiu, Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463. doi: 10.1007/s002110050153.
[15]	B. Leimkuhler and S. Reich, Simulating Hamiltonian dynamics, Number 14. Cambridge University Press, 2004.
[16]	R. I. McLachlan, G. R. W. Quispel, et al., Six Lectures on the Geometric Integration of ODEs, Citeseer, 1998.
[17]	J. Moulin Ollagnier, A. Nowicki and J.-M. Strelcyn, Strelcyn, On the non-existence of constants of derivations: The proof of a theorem of Jouanolou and its development, Bull. Sci. Math., 119 (1995), 195-233.
[18]	G. R. W. Quispel, H. W. Capel, et al., Solving ODEs numerically while preserving all first integrals, Phys. Lett. A, 218 (1996), 223-228. doi: 10.1016/0375-9601(96)00403-3.
[19]	G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Physics A: Mathematical and Theoretical, 41 (2008), 045206. doi: 10.1088/1751-8113/41/4/045206.
[20]	M. F. Reusch, L. Ratzan, N. Pomphrey and W. Park, Diagonal Padé approximations for initial value problems, SIAM J. Sci. Statist. Comput., 9 (1988), 829-838. doi: 10.1137/0909055.
[21]	G. Wanner and E. Hairer, Solving Ordinary Differential Equations Ⅱ, Springer Berlin Heidelberg, 1996. doi: 10.1007/978-3-642-05221-7.
[22]	X. Wu, B. Wang and W. Shi, Efficient energy-preserving integrators for oscillatory Hamiltonian systems, J. Comput. Phys., 235 (2013), 587-605. doi: 10.1016/j.jcp.2012.10.015.
[23]	G. Zhong and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A, 133 (1988), 134-139. doi: 10.1016/0375-9601(88)90773-6.