  A novel class of conservative numerical methods for general conservative
Stratonovich stochastic differential equations with multiple invariants is
proposed and analyzed. These methods, which are called modified averaged vector
field methods, are constructed by modifying the averaged vector field methods
to preserve multiple invariants simultaneously. Based on the prior estimate for
high order moments of the modification coefficient, the mean square convergence
order $1$ of proposed methods is proved in the case of commutative noises. In
addition, the effect of quadrature formula on the mean square convergence order
and the preservation of invariants for the modified averaged vector field
methods is considered. Numerical experiments are performed to verify the
theoretical analyses and to show the superiority of the proposed methods in
long time simulation.
