arXiv subjects

Xianxian Cao

Publications and source records attributed to Xianxian Cao.

4 recordsLinked to original sources

arXiv2026

A Linearized and structure-preserving mixed virtual element method for the extended Fisher-Kolmogorov equation

In thsi paper, based on the leap-frog discretization in time and the mixed virtual element discretization in space, we developed a linearized and structure-preserving numerical algorithm. The main contributions of this work lie in that we not only provide a rigorous proof of the energy dissipation property of the fully discrete numerical scheme, but also establish the unconditionally optimal convergence analysis by means of a inverse inequality. The core of the proof lies in the classified discussion of the relationship between $τ$ and $h$. Finally, two numerical examples are provided to validate the correctness of the theoretical analysis as well as the energy dissipation property of the proposed scheme.

math.NA↗

arXiv2026

A semi-implicit DLN Galerkin finite element method for coupled Ginzburg-Landau equations with general nonlinearity

In this paper, based on the two-step discretization scheme proposed by Dahlquist, Liniger and Nevanlinna (DLN), we develop a semi-implicit Galerkin finite element method for solving the coupled generalized Ginzburg-Landau equations. By virtue of a novel analytical technique, the boundedness of the numerical solution in the infinity norm is established, upon which the unconditionally optimal error estimates in the $L^2$ and $H^1$-norms are further derived. Compared with the space-time error splitting technique commonly adopted in the literature, the analytical method proposed in this paper does not require the introduction of an additional temporal discretization system, thus greatly simplifying the theoretical argument. The core point of the argument lies in the skillful application of the inverse inequality and discrete Agmon inequality to analyze the two cases, namely $τ\leq h$ and $τ>h$, respectively. Three numerical examples covering both two- and three-dimensional scenarios are eventually provided for the validation of the theoretical findings.

math.NA↗

arXiv2025

A new analytical technique of the fully implicit Crank-Nicolson discontinuous Galerkin method for the Ginzburg-Landau Model

In this paper, a fully implicit Crank-Nicolson discontinuous Galerkin method is proposed for solving the Ginzburg-Landau equation. By leveraging a novel analytical technique, we rigorously establish the unique solvability of the constructed numerical scheme, as well as its unconditionally optimal error estimates under both the $L^2$-norm and the energy norm. The core of the proof hinges on the $L^2$-norm boundedness of the numerical solution and the refined estimation of the cubic nonlinear term. Finally, two numerical examples are presented to validate the theoretical findings.

math.NA↗

arXiv2025

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

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 and the spatial step size, specifically distinguishing between the two scenarios of tau^2 \leq h^{k+1}$and$τ^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.

math.NA↗