We introduce a new second order in time relaxation-type scheme for approximating solutions of the Schr\"odinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of \cite{Besse, KK} for the nonlinear Schr\"odinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, satisfies discrete versions of the system's conservation laws, and is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology, that demonstrate the effectiveness and robustness of the new scheme.

翻译：我们引入了时间放松类型的新第二顺序, 以近似于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, 而非线性则以非线性Schr\" poisson 等式的放松方式处理。 对于空间分散化, 我们使用标准符合限制元素计划。 由此产生的计划明确了系统保护法的非线性, 满足了离散版本, 并被视为时间上的第二顺序。 我们最后提出一些数字实验, 包括宇宙学的例子, 以显示新计划的有效性和稳健性 。