scholarly journals On the realization of explicit Runge-Kutta schemes preserving quadratic invariants of dynamical systems

2020 ◽  
Vol 28 (4) ◽  
pp. 327-345
Author(s):  
Yu Ying ◽  
Mikhail D. Malykh
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conservation Law ◽  
Numerical Experiments ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Quadratic Invariant ◽  
Quadratic Invariants ◽  
Runge Kutta ◽  
Autonomous Dynamical Systems

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.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

scholarly journals Hereditary low-mode dynamo model

2021 ◽  
pp. 40-47
Author(s):  
Е.А. Казаков
Keyword(s):  
Dynamical System ◽  
Differential Form ◽  
Difference Schemes ◽  
Dynamo Model ◽  
Two Dimensional ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Suppression Mechanism ◽  
Dimensional Dynamical System ◽  
Runge Kutta

В данной статье рассматривается модель динамо в виде двумерной динамической системы в интегро-дифференциальной форме. В модели реализован стабилизирующий генерацию поля механизм обратной связи в виде подавления α-эффекта функционалом сверточного типа от актуальных и предыдущих значений спиральности и энергии. Наличие этого механизма подавления вводит в модель эредитарность (память). Для модели была построена численная схема ввиде совмещение разностных схем для дифференциальной и интегральной части, двухступенчатый неявный методы Рунге-Кутты и метод трапеций соотвественно. Так же были рассмотрены и графически представлены динамические режимы нашей модели. This article discusses a dynamo model in the form of a two-dimensional dynamical system in integro-differential form. The model implements a stabilizing polarization generator in the form of suppression of the a effect of convolutional type functional from current and previous helicity and energy values. The presence of this suppression mechanism introduces hereditarity (memory) into the model. For modeling, a digital scheme was constructed in the form of a combination of difference schemes for the differential and integral parts, a twostep implicit Runge-Kutta method and a trapezium method, respectively. We also reviewed and graphically presented the dynamic modes of our model.

scholarly journals Diagonally Implicit Symplectic Runge-Kutta Methods with High Algebraic and Dispersion Order

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Y. H. Cong ◽  
C. X. Jiang
Keyword(s):  
Numerical Integration ◽  
Hamiltonian Systems ◽  
Numerical Experiments ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Oscillating Solutions ◽  
Dispersion Order ◽  
Algebraic Order ◽  
Runge Kutta

The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. A diagonally implicit symplectic nine-stages Runge-Kutta method with algebraic order 6 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.

A Modified Runge–Kutta Method for Nonlinear Dynamical Systems With Conserved Quantities

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Guang-Da Hu
Keyword(s):  
Dynamical Systems ◽  
Numerical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Optimization Technique ◽  
Computational Effort ◽  
Conserved Quantities ◽  
Kutta Method ◽  
Nonlinear Dynamical ◽  
Runge Kutta Method ◽  
Runge Kutta

In this paper, explicit Runge–Kutta methods are investigated for numerical solutions of nonlinear dynamical systems with conserved quantities. The concept, ε-preserving is introduced to describe the conserved quantities being approximately retained. Then, a modified version of explicit Runge–Kutta methods based on the optimization technique is presented. With respect to the computational effort, the modified Runge–Kutta method is superior to implicit numerical methods in the literature. The order of the modified Runge–Kutta method is the same as the standard Runge–Kutta method, but it is superior in preserving the conserved quantities to the standard one. Numerical experiments are provided to illustrate the effectiveness of the modified Runge–Kutta method.

On the Harmonic Oscillations for the Motion of a Dynamical System

2017 ◽  
Vol 13 (2) ◽  
pp. 4657-4670
Author(s):  
W. S. Amer
Keyword(s):  
Dynamical System ◽  
Numerical Solutions ◽  
Equation Of Motion ◽  
Parameter Method ◽  
Kutta Method ◽  
Small Parameter Method ◽  
Harmonic Oscillations ◽  
Pivot Point ◽  
Runge Kutta Method ◽  
High Consistency

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

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