Abstract
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 Gröbner 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.
Highlights
Department of Algebra and Geometry Kaili University3, Kaiyuan Road, Kaili, 556011, China (received: February 21, 2019; accepted: October 21, 2019)
The Runge–Kutta method is the most popular numerical method for solving of ordinary differential equations (ODE), the development of this method indicates some symbolic problems.Let’s review some results on Runge–Kutta scheme [1]
In practice it is important that there is an infinite number of sets of rational values for Butcher coefficients
Summary
3, Kaiyuan Road, Kaili, 556011, China (received: February 21, 2019; accepted: October 21, 2019). 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. In Sage there are the excellent tools for investigation of algebraic sets, based on Gröbner basis technique. 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. Key words and phrases: Sympletic Runge–Kutta Scheme; Gröbner basis, Sage, Sagemath
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