Symmetries and Local Conservation Laws of Variational Schemes for the Surface Plasmon Polaritons.

The relation between symmetry and local conservation law, known as the Noether's theorem, plays an important role in modern theoretical physics. As a discrete analogue 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 quasi-neutral background show good properties in secular simulations of the surface plasmon polaritons [Q. Chen et al., e-print arXiv:1810.06205]. We show the discrete local conservation laws admitted by these schemes. Based on the gauge symmetry of the discrete action functional, the discrete charge conservation law is realized locally, which is consistent with the discrete Euler-Lagrange equations obtained by the variational schemes. Based on the discrete Euler-Lagrange equations, the discrete local momentum and energy conservation laws are derived directly, which are rigorous in the theory. The preservation of local conservation laws and Lagrangian symplectic structure ensure the numerical scheme is correct in physics.