We introduce a new structure preserving, 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, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.

翻译：我们引入了一种新的结构保存, 在时间宽松类型中第二顺序, 以接近Schr\'odinger- Poisson 系统的解决方案。 更具体地说, 我们使用Crank- Nicolson 方案作为时间阶梯机制, 而非线性则按照非线性 Schr\'odinger 等式的精神, 以放松方式处理非线性。 对于空间离散性, 我们使用标准符合有限元素方案。 由此产生的方案在非线性方面十分明确, 即它要求每个时间步都采用线性系统的解决方案, 并且满足系统质量保护和能源平衡法的离散版本, 用于恒定的meshes 。 这个方案被看成是时间的第二顺序。 我们最后提出一些数字实验, 包括一个宇宙学的例子和一个显示新计划有效性和稳健性的可变时间步骤的例子。