Numerical conservation issues for the stochastic Korteweg–de Vries equation

In this paper, we focus on structure-preserving issues for the numerical solution of the stochastic Korteweg–de Vries equation, via stochastic ϑ-methods. It is well-known that the aforementioned model exhibits invariant laws along its exact dynamics. Here, our goal is to analyze whether such invariant laws are also reproduced along the numerical dynamics provided by stochastic ϑ-methods. Furthermore, we are also interested in rigorously studying the characterization of such invariant laws along numerical solutions of this model, with respect to the growth of the stochasticity parameter ɛ. At this purpose, the so-called ɛ-expansion of the exact solution to the aforementioned equation will be performed. Numerical results confirming the effectiveness of our analysis are also provided.