We construct, analyze, and numerically validate a class of conservative discontinuous Galerkin (DG) schemes for the Schrödinger-Poisson equation. The proposed schemes all shown to conserve both mass and energy. For the semi-discrete DG scheme the optimal L² error estimates are provided. Efficient iterative algorithms are also constructed to solve the second-order implicit time discretization. The presented numerical tests demonstrate the method’s accuracy and robustness, confirming that the conservation properties help to reproduce faithful solutions over long time simulation.

中文翻译：

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用于 Schrödinger-Poisson 方程的质量和能量守恒的 DG 方法

我们为薛定谔-泊松方程构造、分析和数值验证了一类保守的不连续伽辽金 (DG) 方案。提议的方案都显示出质量和能量守恒。对于半离散 DG 方案，提供了最佳L ²误差估计。还构建了高效的迭代算法来解决二阶隐式时间离散化。所提出的数值测试证明了该方法的准确性和稳健性，证实了守恒特性有助于在长时间模拟中重现忠实的解决方案。