复Ginzburg-Landau方程非线性动力学内罚不连续Galerkin方法的稳定性估计 (Stability estimates for Interior Penalty D.G. Methods for the Nonlinear Dynamics of the complex Ginzburg Landau equation)

from arxiv, In this article Stability of complex Landau equation is presented. After a rigorous Stability analysis for three Discontinuous Galerkin schemes, Numerical Results demonstrate which of these three methods is more stable and what happens if we use big enough penalty terms

This study investigates the complex Landau equation, a reaction diffusion system with applications in nonlinear optics and fluid dynamics. The equation's nonlinear imaginary component introduces rich dynamics and significant computational challenges. We address these challenges using Discontinuous Galerkin (DG) finite element methods. A rigorous stability analysis and a comparative study are performed on three distinct DG schemes : Symmetric Interior Penalty Galerkin (SIPG), Nonsymmetric Interior Penalty Galerkin (NIPG), and Incomplete Interior Penalty Galerkin (IIPG). These methods are compared in terms of their stability and computational efficiency. Our numerical analysis and computational results demonstrate that all three discontinuous Galerkin (DG) schemes are stable. However, the Symmetric Interior Penalty Galerkin (SIPG) scheme proves to be the most robust, as its norm remains bounded even in the presence of nonlinear terms a property not shared by the others. A comparison between the Incomplete Interior Penalty Galerkin (IIPG) and Nonsymmetric Interior Penalty Galerkin (NIPG) schemes shows that IIPG has superior stability properties. For high values of the penalty parameter, all methods exhibit similar stability behavior. Our results highlight the suitability of DG methods for simulating complex nonlinear reaction-diffusion systems and provide a practical framework for selecting the most efficient scheme for a given problem.

翻译：本研究探讨复Ginzburg-Landau方程——一种在非线性光学和流体动力学中具有重要应用的反应扩散系统。该方程的非线性虚部引入了丰富的动力学行为并带来显著的计算挑战。我们采用不连续Galerkin（DG）有限元方法应对这些挑战。针对三种不同的DG格式：对称内罚Galerkin（SIPG）、非对称内罚Galerkin（NIPG）和不完全内罚Galerkin（IIPG），进行了严格的稳定性分析与比较研究。从稳定性和计算效率两方面对这些方法进行了对比。数值分析与计算结果表明，所有三种不连续Galerkin（DG）格式均具有稳定性。然而，对称内罚Galerkin（SIPG）格式被证明是最稳健的，其范数即使在非线性项存在时仍保持有界——这一特性是其他格式所不具备的。不完全内罚Galerkin（IIPG）与非对称内罚Galerkin（NIPG）格式的比较显示，IIPG具有更优的稳定性特性。当罚参数取值较大时，所有方法表现出相似的稳定性行为。本研究结果突显了DG方法在模拟复杂非线性反应扩散系统中的适用性，并为针对特定问题选择最高效的数值格式提供了实用框架。