Second‐Order Least‐Squares Fitting Relaxation Method for Ginzburg‐Landau‐Schrödinger Equation

ABSTRACT The Ginzburg‐Landau‐Schrödinger equation (GLSE) can accurately capture the phase transition and behaviors in nonlinear dynamics. It is challenging to develop a numerical scheme that preserves the energy dissipation property while maintaining high computational efficiency. This inspires us to propose a fully discrete scheme that is linear, decoupled, and unconditionally energy‐stable in this article. The time discretization adopts the relaxation method of regularized energy reconstruction (RRER), combined with the Crank‐Nicolson scheme on a staggered time mesh. The spatial discretization employs the symmetric interior penalty discontinuous Galerkin (SIPG) method based on least‐squares fitting (LSF) and patch reconstruction. The fully discrete scheme can achieve second‐order accuracy in time and arbitrary‐order accuracy in the DG‐norm with only one degree of freedom per element. The unconditional energy stability is rigorously proved, and it is demonstrated that the modified energy is a second‐order temporal approximation of the original energy. Numerical examples validate the effectiveness of the algorithm.