Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2219410.22363/2658-4670-2019-27-1-33-41Research ArticleThe symbolic problems associated with Runge-Kutta methods and their solving in SageYingYu<p>postgraduate student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); assistant professor of Department of Algebra and Geometry, Kaili University</p>yingy6165@gmail.comPeoples’ Friendship University of Russia (RUDN University)Kaili University15122019271334120112019Copyright © 2019, Ying Y.2019<p>Runge-Kutta schemes play a very important role in solving ordinary differential equations numerically. At first we want to present the Sage routine for calculation of Butcher matrix, we call it an rk package. We tested our Sage routine in several numerical experiments with standard and symplectic schemes and verified our result by corporation with results of the calculations made by hand.Second, in Sage there are the excellent tools for investigation of algebraic sets, based on Grbner basis technique. As we all known, the choice of parameters in Runge- Kutta scheme is free. By the help of these tools we study the algebraic properties of the manifolds in affine space, coordinates of whose are Butcher coefficients in Runge-Kutta scheme. Results are given both for explicit Runge-Kutta scheme and implicit Runge-Kutta scheme by using our rk package. Examples are carried out to justify our results. All calculation are executed in the computer algebra system Sage.</p>Sympletic Runge-Kutta SchemeGröbner basisSageSagemathСимплетическая схема Рунге-Кутты, базис Гребнера, Sage, Sagemath[E. Hairer, G. Wanner, S. P. Nørsett, Solving ordinary differential equations, 3rd Edition, Vol. 1, Springer, New York, 2008.][The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.4) (2016). URL https://www.sagemath.org][W. Stein, D. Joyner, SAGE: system for algebra and geometry experimentation, ACM SIGSAM Bulletin 39 (2) (2005) 61-64.][A. Y. Golubkov, A. I. Zobnin, S. O. V., Computer algebra in the Sage system. Manual [Komp’yuternaya algebra v sisteme Sage. Uchebnoye posobiye], Bauman Moscow State Technical University, Moscow, 2013, in Russian.][S. I. Khashin, A symbolic-numeric approach to the solution of the Butcher equations, Canadian Applied Mathematics Quarterly 17 (3) (2009) 555-569.][S. I. Khashin, Butcher algebras for Butcher systems, Numerical Algorithms 63 (4) (2013) 679-689. doi:10.1007/s11075-012-9647-x.][J. H. Verner, Explicit Runge-Kutta pairs with lower stage-order, Numerical Algorithms 65 (3) (2014) 555-577. doi:10.1007/s11075-012-9647-x.][E. Hairer, G. Wanner, C. Lubich, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 2nd Edition, Springer, New York, 2000.][J. M. Sanz-Serna, Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more, SIAM REVIEW 58 (1) (2016) 3-33. doi:10.1137/151002769.][M. N. Gevorkyan, J. V. Gladysheva, Symplectic integrators and the problem of wave propagation in layered media, Bulletin of Peoples’ Friendship University of Russia. Series: Mathematics. Information Sciences. Physics (1) (2012) 50-60, in Russian.][Nuan Fang Xu, Zi-Chen Deng, Yan Wang, Kai Zhang, A symplectic Runge-Kutta method for the analysis of the tethered satellite system, Multidiscipline Modeling in Materials and Structures 13 (1) (2017) 26-35. doi:10.1108/MMMS-11-2016-0060.]