Unconditional long-time stability-preserving second-order BDF fully discrete method for fractional Ginzburg-Landau equation

In this paper, we study the long-time stability-preserving properties of the two-step backward differentiation formula (BDF2) fully discrete scheme for the fractional complex Ginzburg-Landau equation. More precisely, we consider the BDF2 time discretization together with a general spatial spectral discretization and show that the numerical scheme can unconditionally preserve the long-time stability in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2$$\end{document}, H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>1$$\end{document}, H1+alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{1+\alpha }$$\end{document} and H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>2$$\end{document} norms with the aid of the discrete uniform Gronwall lemma. As a special case, we obtain the long-time stability-preserving properties of the fully discrete BDF2 spectral method for the standard complex Ginzburg-Landau equation for the first time. Numerical examples are presented to support our theoretical results.

Automated summary

In this paper, we study the long-time stability-preserving properties of the two-step backward differentiation formula (BDF2) fully discrete scheme for the fractional complex Ginzburg-Landau equation. We consider the BDF2 time discretization together with a general spatial spectral discretization and show that the numerical scheme can unconditionally preserve the long-time stability in L2, H1, H1+alpha, and H2 norms with the aid of the discrete uniform Gronwall lemma. As a special case, we obtain the long-time stability-preserving properties of the fully discrete BDF2 spectral method for the standard complex Ginzburg-Landau equation for the first time. Numerical examples are presented to support our theoretical results.