Gauge-invariant Method For The Time-dependent Ginzburg-Landau Equations In Superconductivity

In this paper,we propose and analyze a stabilized semi-implicit Euler gauge-invariant method for numerical solution of the time-dependent Ginzburg-Landau(TDGL)equations in the two-dimensional space.The proposed method uses the well-known gauge-invariant finite difference approximations with staggered variables in a rectangular mesh,and a stabilized semi-implicit Euler discretization for time integration.The resulted fully discrete system leads to two decoupled linear systems at each time step,thus can be efficiently solved.We prove that the proposed method unconditionally preserves the point-wise boundedness of the solution and is also energy-stable.Moreover,the proposed method under the zeroelectric potential gauge is shown to be equivalent to a mass-lumped version of the lowest order rectangular Nedelec edge element approximation and the Lorentz gauge scheme is equivalent to a mass-lumped mixed finite element method,which indicate such approach is even effective for solving the TDGL problems on non-convex domains although solutions are often with low-regularity in these cases.Numerical examples are also presented to demonstrate effectiveness and robustness of the proposed method.This paper is organized as follows:In the first chapter,we mainly introduce the research background and research status,then we introduce the main conclusions and research methods of this paper.In the second chapter,we propose the stabilized semi-implicit Euler gauge-invariant scheme for solving the TDGL equations and its forms under the zero-electric potential gauge and the Lorentz gauge respectively.In the third chapter,we prove that the proposed scheme satisfies the point-wise boundedness unconditionally and the energy stability.Their equivalence to the lowest order rectangular Nedelec edge finite element approximation is discussed in the fourth chapter.In the fifth chapter,various numerical examples on convex and non-convex domains are presented and the results demonstrate that the proposed schemes are effective and provides correct vortex patterns for superconductors in all cases.