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| We introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of a nonlinear ergodic system of stochastic differential equations, independently of the standard weak order of accuracy of the method. We then present a systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure with a high order of accuracy, again independently of the weak order. Special attention is paid to the high order properties of Lie-Trotter splitting methods for Langevin dynamics, in spite of their standard weak order one.\\
References\\
-A. Abdulle, G. Vilmart, and K.C. Zygalakis, High order numerical approximation of the invariant measure of ergodic SDEs, preprint. 2013.\\
-A. Abdulle, G. Vilmart, and K.C. Zygalakis, Long time accuracy of Lie-Trotter splitting methods for second order stochastic dynamics, preprint. 2014. |
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