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

Zhen Guan, Xianxian Cao

TL;DR

The paper tackles solving the 2D complex Ginzburg–Landau equation with cubic nonlinearity on general polygonal meshes using a weighted IMEX discontinuous Galerkin method. It introduces a two-step, second-order time discretization within a SIPG spatial discretization, coupled with an elliptic projection to achieve unconditional $L^2$ convergence, and proves stability under mild time-step constraints. The authors establish an $L^2$-error bound of $O( au^2+h^{k+1})$, supported by generalized DG inverse inequalities and a transfer formula, and validate the theory with numerical tests on diverse mesh types and curved domains. The work offers a flexible, robust framework for nonlinear parabolic–elliptic PDEs on general meshes with rigorous convergence guarantees.

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 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.

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

TL;DR

The paper tackles solving the 2D complex Ginzburg–Landau equation with cubic nonlinearity on general polygonal meshes using a weighted IMEX discontinuous Galerkin method. It introduces a two-step, second-order time discretization within a SIPG spatial discretization, coupled with an elliptic projection to achieve unconditional convergence, and proves stability under mild time-step constraints. The authors establish an -error bound of , supported by generalized DG inverse inequalities and a transfer formula, and validate the theory with numerical tests on diverse mesh types and curved domains. The work offers a flexible, robust framework for nonlinear parabolic–elliptic PDEs on general meshes with rigorous convergence guarantees.

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 -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}τ^2 > h^{k+1}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.

Paper Structure

This paper contains 11 sections, 8 theorems, 96 equations, 5 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Weighted implicit-explicit discontinuous Galerkin formulation
  3. The variational formulation
  4. Discrete settings of discontinuous Galerkin
  5. The fully discrete numerical scheme
  6. Some vital results
  7. Theoretical analysis of the weighted IMEX discontinuous Galerkin method
  8. The stability of the fully discrete numerical solution
  9. The convergence of the fully discrete numerical scheme
  10. Numerical examples
  11. Conclusions

Key Result

Lemma 2.1

(SunSunGao2016GaoSunSun2015) Assume that $\mathcal{V}$ is an complex inner product space equipped with the inner product $(\cdot, \cdot)_{\mathcal{V}}$ and the induced norm $\|\cdot\|_{\mathcal{V}}$. Then, for any $v^0, v^1, \ldots, v^N \in \mathcal{V}$, it holds that where In addition, the following inequality holds

Figures (5)

  • Figure 1: A non-convex partition composed of 64 elements.
  • Figure 2: A Voronoi partition composed of 64 elements.
  • Figure 3: A mixed mesh partition containing concave and convex elements.
  • Figure 4: A Voronoi mesh partition for the unit circle.
  • Figure 5: $L^2$-$\text{norm}$ of the numerical solution for Example \ref{['example2']} with $h=1/30$ and $\tau=1/100$.

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 5 more