Efficient Stochastic Runge-Kutta Methods for Stochastic Differential Equations with Small Noises (Advances in Applied Mathematics and Mechanics, 2018)
New stochastic Runge-Kutta (SRK) methods for solving the Itô andStratonovich stochastic differential equations (SDEs) with small noises are introduced.These SRK methods contain some multiple stochastic integrals simulated easily andhave high global mean-square error accuracy. To simplify the calculation process, thestochastic rooted tree analysis is developed to estimate the local error and the globalmean-square error estimate for a general class of SRK methods is given. Various improvedSRK methods for the Itô or Stratonovich SDEs with non-commutative, commutative,diagonal, scalar, additive or colored small noises are proposed in turn. Finally,the proposed new SRK methods are examined by four test equations and all of thenumerical results show the high efficiency of our methods.