Linear Stability Analysis Of Two Types Of Random Runge-Kutta Methods

Most of stochastic delay differential equations can not be solved analytically so that numerical methods are one of the important ways to investigate the dynamics of its solution.As we know,Runge-Kutta method is usually applied to solving stochastic differential equation.On the other hand,the stability of an explicit numerical scheme is not as good as the implicit one,but the explicit one has better computational efficiency.Further,a kind of explicit Runge-Kutta method based on Chebyshev polynomial shows good stability properties,However,the stability domain of the method will shrink to zero in some places.In addition,have a kind of explicit RungeKutta method based on Legendre polynomial also shows good stability properties and there is no such problem,and its stability domain can be extended by increasing the stage number of the Runge-Kutta method.For stochastic differential equations,This article first study that the explicit stochastic Runge-Kutta Legendre method,Construct two kinds of format which the order of convergence respectively is 1/2 order and 1 order,And analyze that the linear mean square stability of these methods,get the mean square stability criterion conditions,Our theory is verified by numerical examples.Then in view of the stochastic delay differential equations,we will study that the almost sure exponential stability of an explicit stochastic Runge-Kutta-Chebyshev(S-ROCK)method,which is analyzed by applying the techniques based in a discrete semimartingale convergence thorem.