In a recently developed variational discretization scheme for second order initial value problems ( J. Comput. Phys. 498, 112652 (2024) ), it was shown that the Noether charge associated with time translation symmetry is exactly preserved in the interior of the simulated domain. The obtained solution also fulfils the naively discretized equations of motions inside the domain, except for the last two grid points. Here we provide an explanation for the deviations at the boundary as stemming from the Lagrange multipliers used to implement initial and connection conditions. We show explicitly that the Noether charge including the boundary corrections is exactly preserved at its continuum value over the whole simulation domain, including the boundary points.

翻译：在近期针对二阶初值问题发展的变分离散化方案中（J. Comput. Phys. 498, 112652 (2024)），已证明与时间平移对称性相关的诺特荷在模拟域内部被精确保持。除最后两个网格点外，所得解在域内亦满足朴素离散化的运动方程。本文解释了边界处偏差源于用于施加初始条件与连接条件的拉格朗日乘子。我们明确证明，包含边界修正后的诺特荷在整个模拟域（含边界点）上均精确保持为其连续理论值。