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

The crucial role of Lagrange multipliers in a space-time symmetry preserving discretization scheme for IVPs