  In this paper, we summarize the results about the strong convergence rate of
the Ninomiya-Victoir scheme and the stable convergence in law of its normalized
error that we obtained in previous papers. We then recall the properties of the
multilevel Monte Carlo estimators involving this scheme that we introduced and
studied before. Last, we are interested in the error introduced by discretizing
the ordinary differential equations involved in the Ninomiya-Victoir scheme. We
prove that this error converges with strong order 2 when an explicit
Runge-Kutta method with order 4 (resp. 2) is used for the ODEs corresponding to
the Brownian (resp. Stratonovich drift) vector fields. We thus relax the order
5 for the Brownian ODEs needed by Ninomiya and Ninomiya (2009) to obtain the
same order of strong convergence. Moreover, the properties of our multilevel
Monte-Carlo estimators are preserved when these Runge-Kutta methods are used.
