Abstract
We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Grbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.
Highlights
We implement several explicit Runge–Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage
An elliptic oscillator is used as a test example of the presented package
For the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator
Summary
One of most widespread mathematical models is an autonomous system of ordinary differential equations, i.e., the system of the form. The integrals could not be preserved exactly For this reason, many authors try to construct numerical methods for solving the system of differential equations (1) with the preservation of algebraic integrals without the need to solve nonlinear algebraic equations. Many authors try to construct numerical methods for solving the system of differential equations (1) with the preservation of algebraic integrals without the need to solve nonlinear algebraic equations To overcome these difficulties Buono and Mastroserio [10] suggested a method that uses explicit RK schemes for the construction of new finitedifference schemes which exactly preserve invariants. We will call it the Buono method for shorthand These new schemes are not standard RK schemes, but they are usually called an explicit RK scheme preserving invariants [8]. We will call it the Calvo method for shorthand
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