Stability and Convergence analysis of a Crank-Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

TL;DR

This paper proves the convergence and analyzes the stability of a Crank-Nicolson Galerkin numerical scheme for solving the fractional Korteweg-de Vries equation, including convergence rates and numerical validation.

Contribution

It provides a rigorous convergence analysis of a fully discrete scheme for the fractional KdV equation, incorporating local smoothing effects and numerical rate validation.

Findings

The scheme converges strongly in $L^2$ to a weak solution.

Convergence rates are established for sufficiently regular initial data.

Numerical experiments confirm the theoretical convergence rates.

Abstract

In this paper we study the convergence of a fully discrete Crank-Nicolson Galerkin scheme for the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation, which involves the fractional Laplacian and non-linear convection terms. Our proof relies on the Kato type local smoothing effect to estimate the localized $H^{\alpha/2}$-norm of the approximated solution, where $\alpha \in [1,2)$. We demonstrate that the scheme converges strongly in $L^2(0,T;L^2_{loc}(\mathbb{R}))$ to a weak solution of the fractional KdV equation provided the initial data in $L^2(\mathbb{R})$. Assuming the initial data is sufficiently regular, we obtain the rate of convergence for the numerical scheme. Finally, the theoretical convergence rates are justified numerically through various numerical illustrations.