FOURTH ORDER SYMPLECTIC INTEGRATION WITH REDUCED PHASE ERROR

2008 ◽  
Vol 19 (08) ◽  
pp. 1257-1268 ◽  
Author(s):  
HANS VAN DE VYVER
Keyword(s):  
Hamiltonian Systems ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Phase Error ◽  
Phase Lag ◽  
Order Method ◽  
Symplectic Integration ◽  
Algebraic Order ◽  
Fourth Order Method ◽  
Order Four

In this paper we introduce a symplectic explicit RKN method for Hamiltonian systems with periodical solutions. The method has algebraic order four and phase-lag order six at a cost of four function evaluations per step. Numerical experiments show the relevance of the developed algorithm. It is found that the new method is much more efficient than the standard symplectic fourth-order method.

scholarly journals Local Convergence Analysis of an Efficient Fourth Order Weighted-Newton Method under Weak Conditions

2018 ◽  
Vol 56 (1) ◽  
pp. 23-34
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Newton Method ◽  
Fourth Order ◽  
Local Convergence ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
First Derivative ◽  
Fourth Order Method ◽  
Order Four ◽  
Local Convergence Analysis

Abstract Local convergence analysis of a fourth order method considered by Sharma et. al in [19] for solving systems of nonlinear equations. Using conditions on derivatives upto the order five, they proved that the method is of order four. In this study using conditions only on the first derivative, we prove the convergence of the method in [19]. This way we extended the applicability of the method. Numerical example which do not satisfy earlier conditions but satisfy our conditions are presented in this study.

scholarly journals A Zero-Dissipative Phase-Fitted Fourth Order Diagonally Implicit Runge-Kutta-Nyström Method for Solving Oscillatory Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
K. W. Moo ◽  
N. Senu ◽  
F. Ismail ◽  
M. Suleiman
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Variable Coefficient ◽  
Numerical Experiments ◽  
The Other ◽  
Phase Lag ◽  
Oscillatory Solutions ◽  
Second Order Differential Equations ◽  
Algebraic Order ◽  
Runge Kutta

A new diagonally implicit Runge-Kutta-Nyström (DIRKN) method is constructed for solving second order differential equations with oscillatory solutions. The method is originally based on existing DIRKN method derived by Senu et al. which is three-stage and fourth algebraic order. The new derived method has a variable coefficient with phase-lag of order infinity. The numerical experiments are carried out and the results show the efficiency and accuracy of the new method in comparison with the other DIRKN methods in the literature.

Dynamic Analysis of Prey-Predator Model with Harvesting

2020 ◽  
Vol 5 (5) ◽  
pp. 22-27
Author(s):  
Puskar R. Pokhrel ◽  
Bhabani Lamsal
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dynamic Analysis ◽  
Fourth Order ◽  
Model Equation ◽  
Order Method ◽  
Eigen Values ◽  
Predator Model ◽  
The Stability ◽  
Fourth Order Method

Employing the Lotka -Voltera (1926) prey-predator model equation, the system is presented with harvesting effort for both species prey and predator. We analyze the stability of the system of ordinary differential equation after calculating the Eigen values of the system. We include the harvesting term for both species in the model equation, and observe the dynamic analysis of prey-predator populations by including the harvesting efforts on the model equation. We also analyze the population dynamic of the system by varying the harvesting efforts on the system. The model equation are solved numerically by applying Runge - Kutta fourth order method.

scholarly journals A fourth order method for finding a simple root of univariate function

2015 ◽  
Vol 34 (2) ◽  
pp. 197-211
Author(s):  
D. Sbibih ◽  
Abdelhafid Serghini ◽  
A. Tijini ◽  
A. Zidna
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Simple Root ◽  
Newton's Method ◽  
Order Method ◽  
Numerical Examples ◽  
Quadratic Interpolation ◽  
Univariate Function ◽  
Fourth Order Method

In this paper, we describe an iterative method for approximating asimple zero $z$ of a real defined function. This method is aessentially based on the idea to extend Newton's method to be theinverse quadratic interpolation. We prove that for a sufficientlysmooth function $f$ in a neighborhood of $z$ the order of theconvergence is quartic. Using Mathematica with its high precisioncompatibility, we present some numerical examples to confirm thetheoretical results and to compare our method with the others givenin the literature.

A new efficient implicit four-step method with vanished phase-lag and some of its derivatives for the numerical solution of the radial Schr¨odinger equation

2017 ◽  
Vol 8 (1-2) ◽  
pp. 77 ◽  
Author(s):  
Ali Shokri ◽  
Morteza Tahmourasi
Keyword(s):  
Approximate Solution ◽  
Stability Analysis ◽  
Multistep Method ◽  
Phase Lag ◽  
Order Method ◽  
Step Method ◽  
One Dimensional ◽  
First Derivative ◽  
Algebraic Order ◽  
The One

A new four-step implicit linear sixth algebraic order method with vanished phase-lag and its first derivative is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schr¨odinger equation and related problems. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.

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