Symplectic mixed spectral element time domain method for 3-D Schrödinger–Maxwell equations under Lorenz gauge

Abstract

In this work, Hamiltonian variational principle is employed to prove that Schrödinger–Maxwell (SM) equations under Lorenz gauge exhibit a symplectic structure. Based on this, symplectic mixed spectral element time domain method ($\text{S-MSETD}$) for SM equations under Lorenz gauge is proposed. This method is a structure-preserving geometric algorithm that achieves high accuracy, particularly in long-term simulation. Simultaneously, to address the incompatibility issue between the divergence operator acting on the magnetic vector potential $A$ and the edge spectral element method (SEM), an auxiliary variable $p=\nabla \cdot A$ is introduced. This adjustment allows SM equations under Lorenz gauge to be effectively discretized using mixed SEM (MSEM). Finally, the effectiveness of S-MSETD is validated through numerical simulations.