非线性随机延迟微分方程Heun方法的数值稳定性

本文讨论一般非线性随机延迟微分方程Heun方法的数值稳定性,证明了如果问题本身满足零解是均方指数稳定和均方渐近稳定的充分条件,则当方程的漂移项进一步满足一定的条件时,Heun方法是Ms.稳定的,带线性插值的Heun方法是均方指数稳定的和GMS-稳定的理论结果.文末的数值试验进一步验证了所得的相关结论.     

线性随机延迟积分微分方程Euler-Maruyama方法的稳定性

本文研究一类线性随机延迟积分微分方程Euler-Maruyama方法的MS-稳定性.首先,我们讨论方程真解的均方指数稳定性条件.然后,在此假设条件下,证明了带有复合梯形公式的Euler-Maruyama方法是MS-稳定的.最后,数值试验验证了本文的结论.     

非线性变延迟微分方程隐式Euler方法的数值稳定性

在减弱对非线性刚性变延迟微分方程初值问题本身的约束条件的前提下 ,将已有的文献中隐式Euler方法数值稳定性的结论由常延迟的情形推广到了变延迟的情形 ,证明了隐式Euler方法是稳定的     

随机延迟微分方程的Milstein方法的非线性均方稳定性

本文针对一般的非线性随机延迟微分方程,证明了当系统理论解满足均方稳定性条件时,则当方程的漂移和扩散项满足一定的条件时,Milstein方法也是均方稳定的.数学实验进一步验证了我们的结论.     

非线性中立型随机延迟微分方程随机θ方法的稳定性

本文研究非线性中立型随机延迟微分方程随机θ方法的均方稳定性.在方程解析解均方稳定的条件下,证明了如下结论:当θ∈[0,1/2)时,随机θ方法对于适当小的时间步长是均方稳定的;当θ∈[1/2,1]时,随机θ方法对于任意步长都是均方稳定的.数值结果验证了所获结论的正确性.     

非线性泛函微分与泛函方程线性θ-方法的渐近稳定性

将线性θ-方法用于求解一类非线性泛函微分与泛函方程,结果表明:在问题真解渐近稳定的条件下,A-稳定的线性θ-方法(即1/2≤θ≤1)是渐近稳定的.数值试验的结果验证了所获理论的正确性.     

随机时变线性系统的稳定性

利用构造二次型Ｌｙａｐｕｎｏｖ函数和Ｉｔｏ公式研究了一般ｎ维时变线性Ｉｔｏ型随机微分系统的稳定性,给出了二维时变线性系统的三种常见情形的均方指数 稳定或均方渐近稳定的充分判据。     

一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用

针对非线性不可压缩弹性力学问题,本文提出了一种抽象的稳定化方法并将其应用于非线性不可压缩弹性问题上.在该框架中,我们证明了只要连续的混合问题是稳定的,则可以修正任何满足离散inf-sup条件的混合有限元方法使其是稳定的且最优收敛的.我们将这种抽象的稳定化理论框架应用于非线性不可压缩弹性力学问题,给出了稳定性和收敛性理论结论,并通过数值实验验证了该结论.     

牛顿-正则化方法与一类差分方程反问题的求解

在用牛顿迭代法求解非线性算子方程时,总要求非线性算子的导算子是有界可逆的,即线性化方程是适定的.但在实际数值计算中.即使满足这个条件,也可能出现数值不稳定的现象.为了克服这个困难,[1]将牛顿法与求解线性不适定问题的BG方法(平均核方法)结合起来,在每一步迭代中利用BG方法稳定求解.考虑到Tikhonov的正则化方     

非线性非定常系统的稳定性

本文研究一般情形的非线性非定常系统的稳定性,给出了一个广泛而又实用指数稳定性定理。即使是线性时变系统的情形,我们的结果也具一般性,且有应用方便,简捷等优点。     

Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay

In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.     

非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性

本文研究了数值求解非自治随机微分方程的正则Euler-Maruyama分裂（CEMS）方法,该方程的漂移项系数带有刚性且允许超线性增长,扩散项系数满足全局Lipschitz条件.首先,证明了CEMS方法的强收敛性及收敛速度.其次,证明了在适当条件下CEMS方法是均方稳定的.进一步,利用离散半鞅收敛定理,研究了CEMS方法的几乎必然指数稳定性.结果表明,CEMS方法在漂移系数的刚性部分满足单边Lipschitz条件下可保持几乎必然指数稳定性.最后通过数值实验,检验了CEMS方法的有效性并证实了我们的理论结果.     

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which the drift coefficient satisfies the generalized one-sided Lipschitz condition and the diffusion coefficient satisfies the linear growth condition. Since the delay term $t-[t]$ of SDEPCAs is not continuous and differentiable, the variable substitution method is not suitable. To overcome this difficulty, we adopt new techniques to prove the boundedness of the exact solution and the numerical solution. It is proved that the truncated Euler-Maruyama method is strongly convergent to SDEPCAs in the sense of $L^{\bar{q}}(\bar{q}\ge 2)$. We obtain the convergence order with some additional conditions. An example is presented to illustrate the analytical theory.     

线性随机变时滞微分方程指数Euler方法的收敛性和稳定性

文应用指数Euler方法研究了线性随机变时滞微分方程的收敛性和稳定性;首先,证明了指数Euler方法是$\frac{1}{2}$阶均方收敛的;其次,在解析解均方稳定的前提下,通过跟Euler-Maruyama方法比较发现指数Euler方法在大步长下依然保持解析解的均方稳定性;最后,用数值试验验证了收敛和稳定的结果.     

Strong Predictor-Corrector Approximation for Stochastic Delay Differential Equations

This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations (SDDEs) of Itô-type. The method is proved to be mean-square convergent of order min{$1/2, \hat{p}$} under the Lipschitz condition and the linear growth condition, where $\hat{p}$ is the exponent of Hölder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter $p$ the derived method can have a better stability property than more commonly used numerical methods. That is, for some $p$, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters $p$. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.     

滞后型分段连续随机微分方程的稳定性

本文研究了滞后型分段连续随机微分方程的解析稳定性和数值稳定性问题.首先,利用伊藤公式等方法获得了解析解均方稳定的条件,其次,对于包括均方稳定和T-稳定在内的Euler-Maruyama方法的数值稳定性问题,运用不等式技术和随机分析方法获得了一些新的结果,证明了在一定条件下,Euler-Maruyama方法既是均方稳定又是T-稳定的,推广了随机延迟微分方程的数值稳定性结论.     

CONVERGENCE RATE OF THE TRUNCATED EULER-MARUYAMA METHOD FOR NEUTRAL STOCHASTIC DIFFERENTIAL DELAY EQUATIONS WITH MARKOVIAN SWITCHING

The key aim of this paper is to show the strong convergence of the truncated Euler-Maruyama method for neutral stochastic differential delay equations (NSDDEs) with Markovian switching (MS) without the linear growth condition. We present the truncated Euler-Maruyama method of NSDDEs-MS and consider its moment boundedness under the local Lipschitz condition plus Khasminskii-type condition. We also study its strong convergence rates at time $T$ and over a finite interval $[0, T]$. Some numerical examples are given to illustrate the theoretical results.     

Split-step forward methods for stochastic differential equations

In this paper we discuss split-step forward methods for solving Itô stochastic differential equations (SDEs). Eight fully explicit methods, the drifting split-step Euler (DRSSE) method, the diffused split-step Euler (DISSE) method and the three-stage Milstein (TSM 1a-TSM 1f) methods, are constructed based on Euler-Maruyama method and Milstein method, respectively, in this paper. Their order of strong convergence is proved. The analysis of stability shows that the mean-square stability properties of the methods derived in this paper are improved on the original methods. The numerical results show the effectiveness of these methods in the pathwise approximation of Itô SDEs.