scholarly journals Estimation of Longest Stability Interval for a Kind of Explicit Linear Multistep Methods

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
Y. Xu ◽  
J. J. Zhao
Keyword(s):  
Absolute Stability ◽  
Numerical Experiments ◽  
Multistep Methods ◽  
Linear Multistep Methods ◽  
Stability Intervals ◽  
The Stability ◽  
Order Four

The new explicit linear three-order four-step methods with longest interval of absolute stability are proposed. Some numerical experiments are made for comparing different kinds of linear multistep methods. It is shown that the stability intervals of proposed methods can be longer than that of known explicit linear multistep methods.

scholarly journals On the stability of implicit-explicit linear multistep methods

1997 ◽  
Vol 25 (2-3) ◽  
pp. 193-205 ◽  
Author(s):  
J. Frank ◽  
W. Hundsdorfer ◽  
J.G. Verwer
Keyword(s):  
Multistep Methods ◽  
Linear Multistep Methods ◽  
The Stability

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 Linear Multistep Methods for Impulsive Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
X. Liu ◽  
M. H. Song ◽  
M. Z. Liu
Keyword(s):  
Differential Equations ◽  
Numerical Experiments ◽  
Multistep Methods ◽  
Impulsive Differential Equations ◽  
Multistep Method ◽  
Linear Multistep Methods ◽  
Linear Multistep Method ◽  
Convergence And Stability ◽  
Improved Methods ◽  
Bdf Method

This paper deals with the convergence and stability of linear multistep methods for impulsive differential equations. Numerical experiments demonstrate that both the mid-point rule and two-step BDF method are of orderp=0when applied to impulsive differential equations. An improved linear multistep method is proposed. Convergence and stability conditions of the improved methods are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

scholarly journals Exponential mean-square stability properties of stochastic linear multistep methods

2021 ◽  
Vol 47 (4) ◽  
Author(s):  
Evelyn Buckwar ◽  
Raffaele D’Ambrosio
Keyword(s):  
Numerical Experiments ◽  
Multistep Methods ◽  
Multistep Method ◽  
Linear Multistep Methods ◽  
Linear Multistep Method ◽  
Mean Square ◽  
Stability Properties ◽  
Mean Square Stability ◽  
And Diffusion ◽  
Exponential Mean Square Stability

AbstractThe aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand whether a stochastic linear multistep method can provide an analogous behaviour and which restrictions on the employed stepsize should be imposed in order to reproduce the contractive behaviour. Numerical experiments confirming the theoretical analysis are also given.

An Improved Stiffly Stable Method for Direct Integration of Nonlinear Structural Dynamic Equations

1975 ◽  
Vol 42 (2) ◽  
pp. 464-470 ◽  
Author(s):  
K. C. Park
Keyword(s):  
Multistep Methods ◽  
Nonlinear Problems ◽  
New Method ◽  
Linear Multistep Methods ◽  
Stable Method ◽  
Structural Dynamic ◽  
Linear Problems ◽  
The Stability ◽  
Dynamics Problems ◽  
Derivative Information

The behavior of linear multistep methods has been evaluated for application to structural dynamics problems. By examining the local stability of the currently popular methods as applied to nonlinear problems, it is shown that the presence of historical derivatives can cause numerical instability in the nonlinear dynamics even for methods that are unconditionally stable for linear problems. Through an understanding of the stability characteristics of Gear’s two-step and three-step methods, a new method requiring no historical derivative information has been developed. Superiority of the new method for nonlinear problems is indicated by means of comparisons with currently popular methods.

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