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.

Symplectic Partitioned Runge-Kutta and Symplectic Runge-Kutta Methods Generated by 2-Stage RadauIA Method

2013 ◽  
Vol 444-445 ◽  
pp. 633-636
Author(s):  
Jia Bo Tan
Keyword(s):  
Numerical Integration ◽  
Hamiltonian Systems ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

To preserve the symplecticity property, it is natural to require numerical integration of Hamiltonian systems to be symplectic. As a famous numerical integration, it is known that the 2-stage RadauIA method is not symplectic. With the help of symplectic conditions of Runge-Kutta method and partitioned Runge-Kutta method, a symplectic partitioned Runge-Kutta method and a symplectic Runge-Kutta method are constructed on the basis of 2-stage RadauIA method in this paper.

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 Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
N. A. Ahmad ◽  
N. Senu ◽  
F. Ismail
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
First Order ◽  
Runge Kutta Method ◽  
Algebraic Order ◽  
Runge Kutta ◽  
The Stability ◽  
Order Four

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

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.

Sign in / Sign up

Export Citation Format

Share Document