We introduce a new second order in time Besse-type relaxation scheme for approximating solutions of the Schr\"odinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, the standard conforming finite element method for the spatial discretization whilst the nonlinearity is handled by means of a relaxation approach similar to the one introduced by Besse for the nonlinear Schr\"odinger equation \cite{Besse}. We prove that discrete versions of the system's conservation laws hold and we conclude by presenting some numerical experiments, including an example from cosmology, that demonstrate the effectiveness and robustness of the new scheme.

翻译：我们在时间上引入了一个新的第二顺序,用于接近Schr\'odinger-Poisson系统的解决方案。更具体地说,我们使用Crank-Nicolson计划作为时间阶梯机制,空间离散的标准符合限制元素方法,而非线性则通过类似于Besse为非线性Schr\'odinger等式所引入的放松方法来处理。我们证明,该系统的保护法的离散版本是可行的,我们最后通过提供一些数字实验,包括宇宙学的一个实例,表明新方案的有效性和稳健性。