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.

Accuracy and Reliability of Piecewise-Constant Method in Studying the Responses of Nonlinear Dynamic Systems

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Liming Dai ◽  
Xiaojie Wang ◽  
Changping Chen
Keyword(s):  
Numerical Methods ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamic Systems ◽  
Processing Unit ◽  
Kutta Method ◽  
Piecewise Constant ◽  
Runge Kutta Method ◽  
Runge Kutta

Accuracy and reliability of the numerical simulations for nonlinear dynamical systems are investigated with fourth-order Runge–Kutta method and a newly developed piecewise-constant (P-T) method. Nonlinear dynamic systems with external excitations are studied and compared with the two numerical approaches. Semianalytical solutions for the dynamic systems are developed by the P-T approach. With employment of a periodicity-ratio (PR) method, the regions of regular and irregular motions are determined and graphically presented corresponding to the system parameters, for the comparison of accuracy and reliability of the numerical methods considered. Central processing unit (CPU) time executed in the numerical calculations with the two numerical methods are quantitatively investigated and compared under the same computational conditions. Due to its inherent drawbacks, as found in the research, Runge–Kutta method may cause information missing and lead to incorrect conclusions in comparing with the P-T method.

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 Study on Banded Implicit Runge–Kutta Methods for Solving Stiff Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
M. Y. Liu ◽  
L. Zhang ◽  
C. F. Zhang
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Coefficient Matrix ◽  
Kutta Method ◽  
Stiff Differential Equations ◽  
Banded Matrix ◽  
Fully Implicit ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Cost

The implicit Runge–Kutta method with A-stability is suitable for solving stiff differential equations. However, the fully implicit Runge–Kutta method is very expensive in solving large system problems. Although some implicit Runge–Kutta methods can reduce the cost of computation, their accuracy and stability are also adversely affected. Therefore, an effective banded implicit Runge–Kutta method with high accuracy and high stability is proposed, which reduces the computation cost by changing the Jacobian matrix from a full coefficient matrix to a banded matrix. Numerical solutions and results of stiff equations obtained by the methods involved are compared, and the results show that the banded implicit Runge–Kutta method is advantageous to solve large stiff problems and conducive to the development of simulation.

scholarly journals Static Kirchhoff Rods under the Action of External Forces: Integration via Runge-Kutta Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Ademir L. Xavier Jr.
Keyword(s):  
Reference System ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Kutta Method ◽  
Kirchhoff Equations ◽  
Runge Kutta Method ◽  
System Equations ◽  
Kirchhoff Rods ◽  
Runge Kutta ◽  
External Forces

This paper shows how to apply a simple Runge-Kutta algorithm to get solutions of Kirchhoff equations for static filaments subjected to arbitrary external and static forces. This is done by suitably integrating at once Kirchhoff and filament reference system equations under appropriate initial boundary conditions. To show the application of the method, we display several numerical solutions for filaments including cases showing the effect of gravity.

scholarly journals Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

2019 ◽  
Vol 17 ◽  
pp. 147-154
Author(s):  
Abhinandan Chowdhury ◽  
Sammie Clayton ◽  
Mulatu Lemma
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Adaptive Methods ◽  
Integration Step ◽  
Nonlinear Ordinary Differential Equations ◽  
Kutta Method ◽  
Step Size ◽  
Runge Kutta Method ◽  
Runge Kutta

We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered.

scholarly journals Explicit order 3/2 Runge-Kutta method for numerical solutions of stochastic differential equations by using Itô-Taylor expansion

2019 ◽  
Vol 17 (1) ◽  
pp. 1515-1525
Author(s):  
Yazid Alhojilan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Transport Theory ◽  
Numerical Solutions ◽  
Optimal Transport ◽  
Stochastic Integrals ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract This paper aims to present a new pathwise approximation method, which gives approximate solutions of order $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals Iα by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals Iα. Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

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