Weighted implicit-explicit discontinuous Galerkin methods for two-dimensional Ginzburg–Landau equations on general meshes

Zhen Guan; Xianxian Cao; Zhen Guan; Xianxian Cao

doi:10.3934/nhm.2025059

Networks and Heterogeneous Media

2025, Volume 20, Issue 4: 1367-1391. doi: 10.3934/nhm.2025059

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Research article

Weighted implicit-explicit discontinuous Galerkin methods for two-dimensional Ginzburg–Landau equations on general meshes

Zhen Guan ^,,
Xianxian Cao

School of Mathematics and Statistics, Pingdingshan University, Pingdingshan 467000, China

Received: 11 October 2025 Revised: 02 December 2025 Accepted: 09 December 2025 Published: 12 December 2025

In this paper, a second-order linearized discontinuous Galerkin method on general meshes, which treats the backward differentiation formula of order two (BDF2) and Crank–Nicolson schemes as special cases, is proposed for solving the two-dimensional Ginzburg–Landau equations with cubic nonlinearity. By utilizing the discontinuous Galerkin inverse inequality and the mathematical induction method, the unconditionally optimal error estimate in $ L^2 $-norm is obtained. The core of the analysis in this paper resides in the classification and discussion of the relationship between the temporal step size $ \tau $ and the spatial step size $ h $, specifically distinguishing between the two scenarios of $ \tau^2 \leq h^{k+1} $ and $ \tau^2 > h^{k+1} $, where $ k $ denotes the degree of the discrete spatial scheme. Finally, this paper presents two numerical examples involving various grids and polynomial degrees to verify the correctness of the theoretical results.
- Ginzburg–Landau equations,
- discontinuous Galerkin method,
- general meshes,
- discontinuous Galerkin inverse inequality,
- optimal error estimate
Citation: Zhen Guan, Xianxian Cao. Weighted implicit-explicit discontinuous Galerkin methods for two-dimensional Ginzburg–Landau equations on general meshes[J]. Networks and Heterogeneous Media, 2025, 20(4): 1367-1391. doi: 10.3934/nhm.2025059

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Abstract

In this paper, a second-order linearized discontinuous Galerkin method on general meshes, which treats the backward differentiation formula of order two (BDF2) and Crank–Nicolson schemes as special cases, is proposed for solving the two-dimensional Ginzburg–Landau equations with cubic nonlinearity. By utilizing the discontinuous Galerkin inverse inequality and the mathematical induction method, the unconditionally optimal error estimate in $ L^2 $-norm is obtained. The core of the analysis in this paper resides in the classification and discussion of the relationship between the temporal step size $ \tau $ and the spatial step size $ h $, specifically distinguishing between the two scenarios of $ \tau^2 \leq h^{k+1} $ and $ \tau^2 > h^{k+1} $, where $ k $ denotes the degree of the discrete spatial scheme. Finally, this paper presents two numerical examples involving various grids and polynomial degrees to verify the correctness of the theoretical results.

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