Fully discrete finite difference schemes for the Fractional Korteweg-de Vries equation

TL;DR

This paper develops and analyzes fully discrete finite difference schemes for the fractional Korteweg-de Vries equation, ensuring convergence and conservation properties, validated through numerical experiments for different fractional exponents.

Contribution

The paper introduces a discrete fractional Laplacian operator and proves convergence of the scheme to classical solutions, including conservation properties and numerical validation.

Findings

Convergence of the scheme to classical solutions for fractional KdV.

Preservation of conserved quantities by the Crank-Nicolson scheme.

Numerical validation for various fractional exponents.

Abstract

In this paper, we present and analyze fully discrete finite difference schemes designed for solving the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation involving the fractional Laplacian. We design the scheme by introducing the discrete fractional Laplacian operator which is consistent with the continuous operator, and posses certain properties which are instrumental for the convergence analysis. Assuming the initial data (u_0 \in H^{1+\alpha}(\mathbb{R})), where (\alpha \in [1,2)), our study establishes the convergence of the approximate solutions obtained by the fully discrete finite difference schemes to a classical solution of the fractional KdV equation. Theoretical results are validated through several numerical illustrations for various values of fractional exponent $\alpha$. Furthermore, we demonstrate that the Crank-Nicolson finite difference scheme preserves the inherent conserved quantities along with the improved convergence rates.