分段连续型随机微分方程指数Euler方法的稳定性

给出了线性分段连续型随机微分方程指数Euler方法的均方指数稳定性.经典的对稳定性理论分析,通常应用的是Lyapunov泛函理论,然而,应用该方程本身的特点和矩阵范数的定义给出了该方程精确解的均方稳定性.以往对于该方程应用隐式Euler方法得到对于任意步长数值解的均方稳定性,而应用显式Euler方法得到了相同的结果.最后,给出实例验证结论的有效性.     

带分数布朗Brown的随机比例方程半隐式Euler法的数值解(英文)

本文给出并分析了Poisson随机跳测度驱动的带分数Brown运动的随机比例方程半隐式Euler法的数值解,在局部Lipschitz条件下,证明了在均方意义下半隐式Euler数值解收敛到精确解.     

求解隐式迭代方程的多尺度配置法

用多尺度快速配置法求解病态积分方程的隐式迭代方程.在积分算子是扇形紧算子时,该方法得到了离散隐式迭代方程的近似解.采用Morozov偏差原理作为停止准则,并证明了在该准则下隐式迭代正则化方法所得近似解的收敛率.最后,用数值实验证实理论结果和说明数值方法的有效性.     

随机比例方程带线性插值的半隐式Euler方法的均方收敛性

本文研究非线性随机比例方程带线性捅值的半隐式Euler方法的均方收敛性,证明了这类方法是1/2阶均方收敛的.数值试验验证了所获理论结果的正确性.     

非线性控制系统及其θ方法的压缩性

研究非线性控制系统及其θ方法的压缩性,获得了非线性控制系统压缩性以及θ方法保持其压缩性的充分条件.理论结果表明控制系统的隐式Euler方法比其显式Euler方法的压缩性好.数值结果验证了理论分析结果.     

分数Brown运动时变随机种群收获系统数值解的均方散逸性

讨论了一类带分数Brown运动时变随机种群收获系统数值解的均方散逸性.在一定条件下,利用It公式和Bellman-Gronwall-Type引理,研究了方程(1)具有均方散逸性.分别利用带补偿的倒向Euler方法和分步倒向Euler方法讨论数值解的均方散逸性存在的充分条件,并通过数值算例对所给出的结论进行了验证.     

二维非线性Schr(o)dinger方程的辛与多辛格式

对满足周期边界条件的二维非线性Schrödinger方程,运用中心差分对该方程进行空间离散, 得到一个有限维Hamilton系统,然后用隐式Euler中点格式进行时间离散得到其辛格式. 针对该方程的多辛形式, 运用有限体积法离散,得到一种直平行六面体上的中点型多辛格式. 用所构造的辛与多辛格式对二维非线性Schrödinger方程的平面波解和奇异解进行数值模拟,结果验证了所构 造格式的有效性. 最后, 根据计算结果,对两种格式进行了分析和比较.       

基于非结构自适应网格的二维Euler方程数值求解方法研究

提出了一种基于非结构自适应网格的二维Euler方程的数值解法.采用有限体积法进行空间离散,通量计算采用Jamson中心格式,使得它适用于任意多边形计算单元.为了得到定常解,采用一种显式的四步Runge-Kutta迭代方法对时间进行积分.根据流场参数的变化梯度确定加密边,由加密准则进行自适应网格剖分,然后得到分布合理的加密过后的网格.求解二维Euler方程,对NACA0012翼型进行了数值模拟,通过对自适应前后的数值解的对比,说明所建立的方法是正确的.     

非线性变延迟微分方程隐式Euler法的渐近稳定性

本讨论非线性变延迟微分方程隐式Euler法的渐近稳定性。我们证明，在方程真解渐近稳定的条件下，隐式Euler法也是渐近稳定的。     

随机延迟微分方程的全隐式Euler方法

研究随机延迟微分方程数值解具有重要的意义,目前已有显式和半隐式两种数值方法,还没有全隐式的数值方法.本文构造了一种全隐式Euler方法,在该方法中用一些截断的随机变量代替维纳过程增量△W<,n>,接着证明了全隐式方法是1/2阶收敛的并通过数值实验验证了该方法的收敛性.最后,用数值实验表明在某些情况下全隐式方法的稳定性比半隐式方法好一些.     

Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.     

用线方法求解带混合边值条件的抛物方程

研制了分别用显式Euler法、隐式Euler法、Crank-Nicolson格式(梯形方法)求解带第一、第二及混合边值条件的抛物问题的应用软件,通过求解若干抛物问题对该软件作了测试,获得了预期的数值结果,讨论了时间和空间步长的变化对格式计算结果的影响,得到了三种方法的稳定性、收敛精度和计算量.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Wavelets and adaptive grids for the discontinuous Galerkin method

In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method.     

Numerical approximation for singular second order differential equations

We consider numerical approximation of solutions of singular second order differential equations. In particular, we study the backward (or implicit) Euler method. We prove results concerning consistency, global error and stability. We show that the global error is linear with respect to the step size. Numerical results are also given, which demonstrate the linear convergence and we compare the numerical results with known approximations.     

Extrapolation at stiff differential equations

Summary Asymptotic expansions of the global error of numerical methods are well-understood, if the differential equation is non-stiff. This paper is concerned with such expansions for the implicit Euler method, the linearly implicit Euler method and the linearly implicit mid-point rule, when they are applied tostiff differential equations. In this case perturbation terms are present, whose dominant one is given explicitly. This permits us to better understand the behaviour ofextrapolation methods at stiff differential equations. Numerical examples, supporting the theoretical results, are included.     

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.     

Stochastic model for the nonlinear point reactor kinetics equations in the presence Newtonian temperature feedback effects

A stochastic model for the nonlinear point reactor kinetics equations with Newtonian temperature feedback and multi-group of precursor delayed neutrons is presented. This model is a couple of the stiff stochastic nonlinear differential equations. The matrix formula of this stochastic nonlinear model is solved by the analytical exponential technique (AET). This proposed technique is based on the integration factor, Euler’s method and the exponential function of the coefficient matrix. This exponential function is determined via the eigenvalues and corresponding eigenvectors of the coefficient matrix. The mean neutron population of the stochastic nonlinear model in the presence Newtonian temperature feedback and six-groups of delayed neutrons is computed for various cases of the external reactivity. The numerical results of the analytical exponential technique are compared with the results of the Euler–Maruyama method and the deterministic results. This comparison confirms that the AET for stochastic nonlinear model is efficient to study the natural behavior of neutron population in the presence temperature feedback effects and multi-group of precursor delayed neutrons.