Compact Fourth-Order Linearly Conservative Schemes for the Derivative Nonlinear Schrödinger Equation

Abstract

The derivative nonlinear Schrödinger equation is one of the important classes of integrable systems with extensive applications in nonlinear optics. The numerical solution of the dynamic behavior remains a longstanding computational challenge due to its nonlinear term involving the derivative. In this paper, a second-type derivative nonlinear Schrödinger equation is considered numerically. The present numerical framework comprises two distinct temporal discretization strategies: Crank-Nicolson discretization and three-level average discretization. First of all, a linearized Crank-Nicolson-type scheme and the corresponding compact counterpart are derived at length. We then show that both numerical schemes preserve discrete momentum. Next, by means of the cut-off function method, we prove that the convergence rates of both numerical schemes under the discrete

L^{\infty }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^\infty $$\end{document}

-norm are

\mathcal{O}({\tau }^2+h^2)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {O}}(\tau ^2+h^2)$$\end{document}

or

\mathcal{O}({\tau }^2+h^4)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {O}}(\tau ^2+h^4)$$\end{document}

, where h denotes the spatial step size and

\tau

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau $$\end{document}

denotes the temporal step size. In the meantime, we further derive a three-level average linearized scheme and a corresponding compact analogue for comparison. Discrete conservation quantity and error estimate are verified by taking advantage of several numerical examples with one-/two-soliton solutions. Compared with the implicit schemes in the literature, the numerical schemes in this paper are more efficient than those of the fully implicit ones up to date. More interestingly, numerical results collectively show that the Crank-Nicolson-type linearized difference schemes outperform three-level average linearized ones in the long-time simulation.

Keywords

Derivative nonlinear Schrödinger equation / Soliton solution / Cut-off function / Error estimate / Long time simulation / 65M06 / 65M12

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Funding

Zhejiang Provincial Natural Science Foundation of China(LZ23A010007)

RIGHTS & PERMISSIONS

Shanghai University