  The relation between symmetries and local conservation laws, known as
Noether's theorem, plays an important role in modern theoretical physics. As a
discrete analog of the differentiable physical system, a good numerical scheme
should admit the discrete local conservation laws and inherent mathematical
structures. A class of variational schemes constructed for the
hydrodynamic-electrodynamic model of lossless free-electron gas in a
quasi-neutral background shows good properties in secular simulations of
surface plasmon polaritons [Q. Chen et al., Phys. Rev. E 99, 023313 (2019)]. We
show the discrete local conservation laws admitted by these schemes. Based on
the gauge symmetry of the discrete action functional, a discrete charge
conservation law is realized locally, which is consistent with the discrete
Euler-Lagrange equations obtained from the variational schemes. Based on the
discrete Euler-Lagrange equations, discrete local momentum and energy
conservation laws are derived directly, which are rigorous in theory. The
preservation of the discrete local conservation laws and Lagrangian symplectic
structure ensure that the numerical scheme is correct in physics.
