The Study On Mean Square Stability For Split-step Runge-kutta Methods

Recently,the study of Runge-Kutta(RK)methods for stochastic differential equations based on split-step framework have been widely concerned.In fact,RK method is a very important numerical method for solving differential equations.Besides,there are several sorts of common implicit RK methods have very good stability when solving differential equations.Therefore,the study of the stability for stochastic differential equations via RK methods is favored by more and more scholars.What`s more,in order to improve the stability or stable area of stochastic RK methods,all kinds of different schemes emerge in endlessly.In this paper,we will study a kind of universal split-step RK(SSRK)methods,and give sufficient conditions by means of mean square for stability of stochastic differential equations.It is worth noting that this paper gives a conclusion,which can directly determine the stability of the RK methods by using the stability functions of the RK methods.Main content of this work are as follows.Firstly,the mean square stability of SSRK methods for linear stochastic differential equations is studied,and the sufficient condition is given.At the same time,based on the above sufficient conditions,the stability of several kinds of common RK methods for solving stochastic differential equations are given.Besides,the corresponding proofs are also derived.The numerical example verifies the effectiveness of theoretical results.Secondly,considering the stability of SSRK methods are proposed to improve.In this paper,we will give the concrete manifestation of the improved SSRK methods,and obtain the sufficient conditions of the stability for the improved SSRK methods in the mean square.In the same way,based on the above sufficient conditions,the stability of several kinds of common RK methods for stochastic differential equations are given by means of mean square and corresponding proofs follow.Numerical examples demonstrate the validity of the theoretical results.Finally,with the general application of nonlinear stochastic differential equations,based on the improved SSRK methods,the mean square stability of the nonlinear stochastic differential equations is also be discussed.Similarly,the sufficient conditions of the stability for improved SSRK methods are obtained.The numerical example verifies the theoretical outcome is effective.