Efficient Stochastic Runge-Kutta Methods

This thesis focuses on the efficient stochastic Runge-Kutta(SRK)methods for stochastic differential equations(SDEs).Various new SRK methods are constructed based on some new techniques.What's more,the theoretical analysis and the numerical results show that the new methods are efficient.In chapter 1,some background and progress on the SDEs and its related numerical methods are introduced.Moreover,a general class of SRK methods and its order conditions are reviewed.Finally,we make an outline about the main work of this thesis.In chapter 2,based on some new techniques,various new SRK meth-ods with high strong order are constructed for the Stratonovich SDEs.Compared with existing well-known SRK methods,the number of order conditions of the new SRK methods is less,and their form is simpler."What's more,the numerical results show that the new methods are also more advantageous in terms of computational efficiency.A key problem in the implementation of the methods with high strong order is how to calculate efficiently the multiple stochastic inte-grals.In chapter 3,an explicit construction of the optimal approximation(in the mean-square sense)to the stochastic integrals used in some strong second-order methods is proposed based on a Karhunen-Loeve expansion of a Wiener process.The optimal approximation is more efficient by comparison with the stochastic Fourier series approximation and the s-tochastic Taylor approximation.The numerical results show that the actual calculation efficiency of the strong second-order methods is higher than the methods with strong order 1.5.In chapter 4,based on the stochastic rooted tree analysis,the global mean-square error estimate of a general class of SRK methods is given for the SDEs with small noises.Various SRK methods with high precision are proposed by using this global mean-square error estimate.It is worth mentioning that those multiple stochastic integrals simulated difficultly are not necessary for these high precision SRK methods.The numerical results show that the calculation efficiency of these high precision SRK methods is higher than existing SRK methods that are suitable for the SDEs with small noises.In chapter 5,a new class of weak second-order SRK methods for Ito SDEs with an m-dimensional Wiener process is introduced.Compared with existing well-known weak second-order SRK methods,the order con-ditions of the new methods are simpler,and the computation costs are less.In chapter 6,for stiff SDEs,the new explicit stabilized SRK methods with strong or weak order are obtained by combining the second order orthogonal Runge-Kutta-Chebyshev methods with the new methods in chapters 2 and 5.Compared with existing explicit stabilized SRK meth-ods based on Chebyshev polynomials,the new methods have advantages in terms of both stability and computational cost.