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 method to preserve multiple invariants simultaneously. Based on the a prior estimate for high-order moments of the modification coefficient, the mean square convergence order 1 of the proposed methods is proved in the case of commutative noises. In addition, the effect of the quadrature formula on the mean square convergence order and the preservation of invariants for 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 the long time simulation.

中文翻译：

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为保守随机微分方程保留多个不变量的修正平均向量场方法

提出并分析了一类具有多重不变量的一般保守Stratonovich随机微分方程的新保守数值方法。这些方法被称为修正平均向量场方法，是通过修正平均向量场方法来同时保持多个不变量而构建的。基于修正系数高阶矩的先验估计，证明了所提方法在交换噪声情况下的均方收敛阶数为1。此外，还考虑了求积公式对均方收敛阶数的影响以及修正平均矢量场方法的不变量保持。