Mechanical Systems With Impacts: Comparison of Several Numerical Methods

1999 ◽  
Author(s):  
C. H. Lamarque ◽  
O. Janin
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Infinite Number ◽  
Mechanical Systems ◽  
Numerical Results ◽  
Degree Of Freedom ◽  
Periodic Behavior ◽  
Numerical Tests ◽  
Runge Kutta ◽  
Theoretical Results

Abstract We study the performances of several numerical methods (Paoli-Schatzman, Newmark, Runge-Kutta) in order to compute periodic behavior of a simple one-degree-of-freedom impacting oscillator. Some theoretical results are given and numerical tests are performed. We compare mathematical and numerical results using our simple example exhibiting either finite or infinite number of impacts per period. Comparison of exact and numerical solution provides a practical order for each scheme. We conclude about the use of the different numerical methods.

scholarly journals A Class of New Pouzet-Runge-Kutta-Type Methods for Nonlinear Functional Integro-Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Chengjian Zhang
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Numerical Stability ◽  
Numerical Experiments ◽  
Integrodifferential Equations ◽  
Stability Criteria ◽  
Nonlinear Functional ◽  
Runge Kutta ◽  
Theoretical Results

This paper presents a class of new numerical methods for nonlinear functional-integrodifferential equations, which are derived by an adaptation of Pouzet-Runge-Kutta methods originally introduced for standard Volterra integrodifferential equations. Based on the nonclassical Lipschitz condition, analytical and numerical stability is studied and some novel stability criteria are obtained. Numerical experiments further illustrate the theoretical results and the effectiveness of the methods. In the end, a comparison between the presented methods and the existed related methods is given.

An Interpolation-Based Method for Multidimensional Extrapolation

2014 ◽  
Vol 1079-1080 ◽  
pp. 654-659
Author(s):  
Ru Chao Shi ◽  
Yong Chi Li
Keyword(s):  
Numerical Results ◽  
Numerical Implementation ◽  
Order Accuracy ◽  
First Order ◽  
Numerical Tests ◽  
Pde Method ◽  
Theoretical Results

This paper presents an interpolation-based method for multidimensional extrapolation. A series of interpolation formulations are proposed to extrapolate functions normal to the interface between two regions. Theoretical proofs and relevant analysis are also presented. The method developed maintains the characteristics of implicit interface. The interface inside every cell is treated smoothly by assuming that curvature is equal everywhere. The method developed and numerical results are verified by comparing to the results by PDE method and theoretical results. Numerical tests demonstrate that the method developed is first-order accuracy and also more efficient in numerical implementation and more accurate than PDE method.

scholarly journals Stability Analysis of One-Leg Methods for Nonlinear Neutral Delay Integrodifferential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Yuexin Yu ◽  
Liping Wen
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Analytical Solution ◽  
Global Stability ◽  
Numerical Solution ◽  
Integrodifferential Equations ◽  
Neutral Delay ◽  
Numerical Tests ◽  
Theoretical Results

This paper is concerned with the numerical solution of nonlinear neutral delay integrodifferential equations (NDIDEs). The adaptation of one-leg methods is considered. It is proved that anA-stable one-leg method can preserve the global stability and a stronglyA-stable one-leg method can preserve the asymptotic stability of the analytical solution of nonlinear NDIDEs. Numerical tests are given to confirm the theoretical results.

NUMERICAL ANALYSIS OF DEGENERATE CONNECTING ORBITS FOR MAPS

2004 ◽  
Vol 14 (10) ◽  
pp. 3385-3407 ◽  
Author(s):  
WOLF-JÜRGEN BEYN ◽  
THORSTEN HÜLS ◽  
YONGKUI ZOU
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Numerical Methods ◽  
Error Analysis ◽  
Error Estimates ◽  
Discrete Dynamical Systems ◽  
Numerical Results ◽  
Connecting Orbits ◽  
Theoretical Results ◽  
Degenerate Cases

This paper contains a survey of numerical methods for connecting orbits in discrete dynamical systems. Special emphasis is put on degenerate cases where either the orbit loses transversality or one of its endpoints loses hyperbolicity. Numerical methods that approximate the connecting orbits by finite orbit sequences are described in detail and theoretical results on the error analysis are provided. For most of the degenerate cases we present examples and numerical results that illustrate the applicability of the methods and the validity of the error estimates.

scholarly journals A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

2020 ◽  
Vol 17 (1) ◽  
pp. 0166
Author(s):  
Hussain Et al.
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

scholarly journals Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Yanping Yang ◽  
Yonglei Fang ◽  
Xiong You ◽  
Bin Wang
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Results ◽  
First Order ◽  
Truncation Errors ◽  
New Methods ◽  
Exponentially Fitted ◽  
Runge Kutta ◽  
First Order Differential Equations

The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods.

A Meshless Method for the Numerical Solution of the Generalized Burgers Equation

2011 ◽  
Vol 101-102 ◽  
pp. 275-278
Author(s):  
Feng Xin Sun ◽  
Ju Feng Wang
Keyword(s):  
Numerical Methods ◽  
Exact Solutions ◽  
Numerical Solution ◽  
Meshless Method ◽  
Burgers Equation ◽  
Generalized Equation ◽  
Numerical Results ◽  
Discrete Scheme ◽  
Generalized Burgers Equation ◽  
Mls Approximation

Based on the MLS approximation, a meshless method for the numerical solution of the generalized Burger’s equation is presented in this paper. The nonlinear discrete scheme of the generalized equation is obtained, and is solved with the method of iteration. Compared with numerical methods based on mesh, the meshless method needs only the scattered nodes instead of meshing the domain of the problem. An example is given to demonstrate the accuracy of the proposed method. The numerical results agree well with the exact solutions.

scholarly journals Some Numerical Results of Multipoints Bomndary Value Problems Arise in Environmental Protection

2016 ◽  
Vol 68 (1) ◽  
pp. 50-52
Author(s):  
Daniel N. Pop
Keyword(s):  
Numerical Methods ◽  
Environmental Protection ◽  
Pollutant Transport ◽  
Approximate Solutions ◽  
Numerical Results ◽  
B Splines ◽  
Runge Kutta

Abstract In this paper, we investigate two problems arise in pollutant transport in rivers, and we give some numerical results to approximate this solutions. We determined the approximate solutions using two numerical methods: 1. B-splines combined with Runge-Kutta methods, 2. BVP4C solver of MATLAB and then we compare the run-times.

scholarly journals Resolução Numérica da equação diferencial de resfriamento de newton pelos métodos de EULER e RUNGE-KUTTA

2016 ◽  
Vol 2 (1) ◽  
pp. 10-25
Author(s):  
Andresa Pescador ◽  
Zilmara Raupp Quadros Oliveira
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Euler Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Analytical Resolution ◽  
Important Branch ◽  
First Order Differential Equations

This article presents the first-order differential equations, which are a very important branch of mathematics as they have a wide applicability, in mathematics, as in physics, biology and economy. The objective of this study was to analyze the resolution of the equation that defines the cooling Newton's law. Verify its behavior using some applications that can be used in the classroom as an auxiliary instrument to the teacher in addressing these contents bringing answers to the questions of the students and motivating them to build their knowledge. It attempted to its resolution through two numerical methods, Euler method and Runge -Kutta method. Finally, there was a comparison of the approach of the solution given by the numerical solution with the analytical resolution whose solution is accurate.

Numerical Solution of Volterra Integro-Differential Equations Using Improved Runge-Kutta Methods

2019 ◽  
Vol 892 ◽  
pp. 193-199
Author(s):  
Faranak Rabiei ◽  
Fatin Abd Hamid ◽  
Nafsiah Md Lazim ◽  
Fudziah Ismail ◽  
Zanariah Abdul Majid
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Numerical Solution ◽  
Quadrature Rule ◽  
Numerical Results ◽  
Test Problems ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Interpolation Polynomials

In this paper, we proposed the numerical solution of Volterra integro-differential equations of the second kind using Improved Runge-Kutta method of order three and four with 2 stages and 4 stages, respectively. The improved Runge-kutta method is considered as two-step numerical method for solving the ordinary differential equation part and the integral operator in Volterra integro-differential equation is approximated using quadrature rule and Lagrange interpolation polynomials. To illustrate the efficiency of proposed methods, the test problems are carried out and the numerical results are compared with existing third and fourth order classical Runge-Kutta method with 3 and 4 stages, respectively. The numerical results showed that the Improved Runge-Kutta method by achieving the higher accuracy performed better results than existing methods.

Sign in / Sign up

Export Citation Format

Share Document