Symplectic Mixed Spectral Element Time-Domain Method for 3-D Schrödinger–Maxwell Equations With Generalized Coulomb Gauge

In this work, the variational principle of Hamilton is applied to construct constrained Hamiltonian systems for Schrödinger–Maxwell (SM) equations with generalized Coulomb gauge. Then, on the foundation of constrained Hamiltonian systems, symplectic mixed spectral element time-domain (S-MSETD) method for 3-D SM equations is proposed to guarantee the zero divergence of the magnetic vector potential (A) in edge spectral element method (SEM) and control both the probability and energy error for all time steps. In S-MSETD of SM equations, mixed SEM (MSEM) is employed for the spatial discretization of the wave function (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Psi $ </tex-math></inline-formula>) and A under generalized Coulomb gauge, and the time-stepping scheme is the symplectic implicit midpoint (IM) method. Several numerical examples are given to verify that S-MSETD maintains high accuracy after the long-term simulation.