Stochastic Newmark scheme

In this paper, we are concerned with numerical resolution of the dynamical equation of linear oscillators. An answer is given to the following question: can one use widely implemented Newmarkʹs algorithm when the acceleration is a white noise instead of a regular deterministic one? After showing drawbacks of such a practice, a modification of the usual Newmark schemes is introduced, to be used for the numerical resolution of such a stochastic dynamical equation. The modification is only in the stochastic part, so that already implemented softwares can be used, the input only having to be carefully adapted. After that, mathematical analysis of this new scheme is fulfilled. First and second moments, as well as the power spectral density resulting from the use of this scheme are compared with the corresponding quantities resulting from the theoretical process, and also from usual Newmark scheme and Euler schemes. Almost sure convergence of this new scheme is proved.