A numerical method for the fractional Zakharov-Kuznetsov equation

TL;DR

This paper introduces a spectral Galerkin method for solving the fractional Zakharov-Kuznetsov equation, providing convergence proofs, error analysis, and numerical validation for nonlocal dispersive wave modeling.

Contribution

It develops a fully discrete Fourier spectral Galerkin scheme for the fractional ZK equation, including convergence analysis and an efficient time discretization method.

Findings

Spectral convergence order $ ext{O}(N^{-r})$ for initial data in $H^r$ with $r \\geq \\alpha+1$

Exponential convergence for analytic solutions

Numerical experiments confirm theoretical accuracy and efficiency

Abstract

This paper develops a fully discrete Fourier spectral Galerkin (FSG) method for the fractional Zakharov-Kuznetsov (fZK) equation posed on a two-dimensional periodic domain. The equation generalizes the classical ZK model to incorporate nonlocal dispersion through a fractional Laplacian of order $\alpha \in [1,2]$. We first propose a semi-discrete FSG scheme in space that preserves the discrete analogs of mass, momentum, and energy. The existence and uniqueness of semi-discrete solutions are established. Using compactness arguments, we prove the uniform convergence of the semi-discrete approximations to the unique solution of the fZK equation for the periodic initial data in $H^{1+\alpha}_{\mathrm{per}}(\Omega)$. The method achieves spectral convergence of order $\mathcal{O}(N^{-r})$ for initial data in $H^r_{\mathrm{per}}$ with $r \geq \alpha+1$, and exponential convergence for analytic solutions utilizing a modified projection. An efficient integrating-factor Runge-Kutta time discretization is designed to handle the stiff fractional term, and an error analysis is presented. Numerical experiments validate the theoretical results and demonstrate the method's effectiveness across various fractional orders.