两类随机Runge-Kutta方法的线性稳定性分析

发布时间：2018-03-06 19:27

本文选题：随机微分方程　切入点：Runge-Kutta　出处：《上海师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：随机微分方程的解析解一般难以求得,因此数值方法成为研究随机微分方程解的行为的主要工具之一,其中龙格库塔(Runge-Kutta)方法是求解随机微分方程的重要方法之一.另一方面,显格式的数值稳定性虽不如隐格式,但显格式具有更高的计算效率.其中,有一类基于Chebyshev多项式构造的显式Runge-Kutta方法具有良好的稳定性性质,但在某些位置该方法稳定域的宽度会缩小为零,而基于Legendre多项式构造的显式Runge-Kutta方法不存在这样的问题.这两种方法的数值稳定区域都随着Runge-Kutta方法阶段数的增加而扩大.针对随机微分方程,本文首先考虑显式随机Runge-Kutta Legendre方法,构造收敛阶分别为1/2阶与1阶的格式,并且分析了这些方法的线性均方稳定性,得到均方稳定性判别条件,并通过数值例子验证了我们的理论结果.然后针对随机时滞微分方程,我们运用离散半秧收敛定理分析了显式随机Runge-Kutta Chebyshev方法的几乎处处指数稳定性.
[Abstract]:The analytical solutions of stochastic differential equations are generally difficult to obtain, so numerical methods have become one of the main tools to study the behavior of stochastic differential equations, among which Runge-Kuttamethod is one of the important methods for solving stochastic differential equations. Although the numerical stability of explicit schemes is not as good as implicit schemes, explicit schemes have higher computational efficiency. Among them, a class of explicit Runge-Kutta methods based on Chebyshev polynomials have good stability. But at some point, the width of the stability region of the method shrinks to zero, However, the explicit Runge-Kutta method based on Legendre polynomials does not have such a problem. The numerical stability region of these two methods expands with the increase of the number of stages in the Runge-Kutta method. For stochastic differential equations, the explicit stochastic Runge-Kutta Legendre method is first considered in this paper. The schemes with convergence order of order 1/2 and order 1 are constructed, and the linear mean square stability of these methods is analyzed, and the criterion conditions of mean square stability are obtained. Numerical examples are given to verify our theoretical results. Then, for stochastic delay differential equations, we use the discrete semi-convergence theorem to analyze the almost everywhere exponential stability of explicit stochastic Runge-Kutta Chebyshev method.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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