A Relaxed Energy-Optimized Exponential Scalar Auxiliary Variable Method for Ginzburg-Landau Equation and Applications in Vortex Dynamics

An energy-optimized exponential scalar auxiliary variable (EOP-ESAV) method for the Ginzburg-Landau equation (GLE) with a relaxation step in exponential scalar auxiliary variable (ESAV) schemes is proposed. The EOP-ESAV method is an explicit linear approach that allows for high-order, computationally efficient schemes. Furthermore, this approach retains the convergence order of the ESAV schemes while achieving smaller numerical errors. Most notably, we rigorously prove that the proposed method maintains a modified energy dissipation law that optimally approximates the original energy within the relaxation framework. Extensive numerical experiments are conducted to demonstrate notable improvements of the EOP-ESAV method in accuracy and numerical stability. The EOP-ESAV schemes are also applied on the simulation of vortex dynamics in GLEs with various setups, studying vortex dipole interactions and lattice motion patterns.