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

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.