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.

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.

DYNAMICS OF LINEAR MULTISTEP METHODS FOR DELAY DIFFERENTIAL EQUATIONS

2004 ◽  
Vol 14 (01) ◽  
pp. 329-336
Author(s):  
HONGJIONG TIAN ◽  
QIAN GUO
Keyword(s):  
Multistep Methods ◽  
Multistep Method ◽  
Linear Multistep Methods ◽  
Linear Multistep Method ◽  
Step Size ◽  
Time Step ◽  
True Solution ◽  
Delay Differential ◽  
Stable Linear ◽  
Period Two Solutions

In this paper we study the relationship between the asymptotic behavior of a numerical simulation by linear multistep method and that of the true solution itself for fixed step sizes. The numerical method is viewed as a dynamical system in which the step size acts as a parameter. Numerical stability of linear multistep method for nonlinear delay differential equation is investigated and we prove that A-stable linear multistep methods are NP-stable. It is shown that a consistent zero-stable linear multistep method does not admit spurious fixed points. The existence of spurious period-two solutions in the time-step is also studied. Finally we give a simple example to illustrate instability of the spurious period-two solutions.

scholarly journals Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 297-313 ◽  
Author(s):  
Desmond J. Higham ◽  
Xuerong Mao ◽  
Andrew M. Stuart
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Finite Time ◽  
Mean Square ◽  
Convergence Conditions ◽  
Finite Time Convergence ◽  
Mean Square Stability ◽  
Exponential Mean Square Stability ◽  
Positive Results

AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

EXPONENTIAL MEAN SQUARE STABILITY OF STOCHASTICALLY FORCED INVARIANT MANIFOLDS FOR NONLINEAR SDEs

2007 ◽  
Vol 07 (03) ◽  
pp. 389-401 ◽  
Author(s):  
L. B. RYASHKO
Keyword(s):  
Linear Extension ◽  
Invariant Manifolds ◽  
Function Technique ◽  
Nonlinear Stochastic System ◽  
General Notion ◽  
Mean Square ◽  
Extension System ◽  
Mean Square Stability ◽  
The Stability ◽  
Exponential Mean Square Stability

An exponential mean square stability for the invariant manifold [Formula: see text] of a nonlinear stochastic system is considered. The stability analysis is based on the [Formula: see text]-quadratic Lyapunov function technique. The local dynamics of the nonlinear system near manifold is described by the stochastic linear extension system. We propose a general notion of the projective stability (P-stability) and prove the following theorem. The smooth compact manifold [Formula: see text] is exponentially mean square stable if and only if the corresponding stochastic linear extension system is P-stable.

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.

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