Stable implicit numerical algorithm of time-dependent Ginzburg-Landau theory coupled with thermal effect for vortex behaviors in hybrid superconductor systems

Hybrid multi-superconducting structures exist in a variety of superconducting devices, such as superconductor-insulator-superconductor multilayer structure in superconducting radio-frequency cavities, bilayer structures in superconducting electronic devices, and superconducting wires. Investigating the vortex dynamics at microscopic scale is crucial for applications of hybrid superconducting structures.The time-dependent Ginzburg-Landau (TDGL) theory is a powerful tool for describing the vortex dynamics in superconductors through the order parameter psi and vector potential A. However, the difference in order parameter psi, coherence length xi, and GL parameters kappa among the components of hybrid systems will bring significant challenges to numerical simulation of TDGL equations. Meanwhile, the energy dissipation associated with vortex motion necessitates considering the thermal effects on vortex dynamics. In this paper, we introduce an efficient, stable, and parallel implicit finite-difference algorithm, implemented on GPU, for coupling the TDGL and thermal diffusion equations for hybrid structures. Linearization of nonlinear source terms is applied to TDGL-II to enhance the stability of algorithm. The iterative Jacobi method is applied to the generalized TDGL-I. Alternating direction implicit methods combined with tridiagonal matrix method or CTDMA are used to solve TDGL-II and heat diffusion equations with different boundary conditions. This algorithm enables us to explore the vortex dynamics with associated thermal effects of mesoscopic large hybrid multi-superconducting structures within reasonable amounts of computational time. Our approach aids in revealing and understanding the underlying physical mechanisms behind the collective response of vortices, and contributes to the mastery, adjustment, and optimization of superconductivities in hybrid structures.