Analysis of a Fully Discrete Local Discontinuous Galerkin Scheme for the Fractional Korteweg-de Vries Equation

Abstract

We develop and analyze a fully discrete local discontinuous Galerkin (LDG) method for the fractional Korteweg-de Vries (KdV) equation, where the nonlocal dispersion is modeled by a fractional Laplacian with exponent

\alpha \in (1,2)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (1,2)$$\end{document}

in both one and multiple space dimensions. The nonlocal nature of this operator introduces singularities and analytical challenges, which we overcome using tools from fractional calculus and appropriate regularity estimates. By reformulating the fractional Laplacian as a composition of first-order derivatives and a fractional integral, we design a stable LDG scheme with carefully constructed numerical fluxes at element interfaces and boundaries. We rigorously prove the

L^2

\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}

-stability of the semi-discrete scheme and establish an a priori error estimate of order

\mathcal{O}(h^{k+1})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}(h^{k+1})$$\end{document}

for linear fluxes and

\mathcal{O}(h^{k+\frac{1}{2}})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}(h^{k+\frac{1}{2}})$$\end{document}

for general nonlinear fluxes. The stability and convergence analysis naturally extends to higher-dimensional problems, highlighting the adaptability of the proposed framework. For time discretization, we apply the Crank-Nicolson (CN) method to obtain a fully discrete scheme and demonstrate similar stability and convergence properties. Numerical experiments support the theoretical findings and confirm that the scheme achieves optimal convergence rates, illustrating both its accuracy and computational efficiency.

Keywords

Fractional Korteweg-de Vries (KdV) equation / Fractional Laplacian / Fractional Sobolev spaces / Local discontinuous Galerkin (LDG) method / Primary: 65M60 / 35Q53 / Secondary: 65M12

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