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October 8, 2025

A Linearly Implicit and Structure‐Preserving Scheme for the Ginzburg–Landau–Schrödinger Equation

DZDan ZhaoUniversity of Electronic Science and Technology of China YTYanbin TangHuazhong University of Science and Technology DLDongfang LiKunming University of Science and Technology

Key Points

The proposed numerical scheme maintains unconditional energy stability and accuracy in simulations.
It achieves second-order accuracy in the temporal direction and high accuracy in space with FEM.
Testing confirms that the method is mass- and energy-conserving, particularly for the nonlinear Schrödinger equation.
Error estimates remain optimal regardless of the ratio of temporal and spatial step sizes.

Abstract

ABSTRACT A fully discrete numerical scheme is proposed for solving the Ginzburg–Landau–Schrödinger equation (GLSE). The spatial discretization is performed by using the finite element method (FEM), and the temporal discretization is done by using the modified leap‐frog method. The fully discrete scheme is linearly implicit. It is proven that the fully discrete scheme is unconditionally energy‐stable. In particular, when the parameter , the GLSE reduces to the nonlinear Schrödinger equation. In this case, the proposed scheme is both mass‐ and energy‐conserving. It is shown that the scheme is of second‐order accuracy in the temporal direction and of ‐th order accuracy in the space direction with ‐degree FEM. The optimal error estimate holds without any restrictions on the ratio of temporal‐spatial step‐sizes. Several numerical examples are presented to confirm the theoretical results.

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