New stochastic Runge-Kutta (SRK) methods for solving the Itô and
Stratonovich stochastic differential equations (SDEs) with small noises are introduced.
These SRK methods contain some multiple stochastic integrals simulated easily and
have high global mean-square error accuracy. To simplify the calculation process, the
stochastic rooted tree analysis is developed to estimate the local error and the global
mean-square error estimate for a general class of SRK methods is given. Various improved
SRK 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 the
numerical results show the high efficiency of our methods.