A numerical method for the fractional Zakharov-Kuznetsov equation

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 by replacing the Laplacian with a fractional Laplacian of order \(α\in(0,2]\), thereby covering the classical ZK equation \(α=2\), the higher-dimensional Benjamin--Ono--ZK equation \(α=1\), and weaker fractional-dispersion regimes \(0<α<1\). We first propose a semi-discrete FSG scheme in space that preserves the discrete analogues of mass, momentum, and Hamiltonian energy. Using periodic Kato--Ponce product and commutator estimates, we prove local-in-time uniform Sobolev bounds and strong convergence of the semi-discrete approximations to the unique strong solution in \(C([0,\bar T];L^2_{\mathrm{per}}(Ω))\), for the initial condition in \(H^s_{\mathrm{per}}(Ω)\), \(s\geq 2+α\), and, as by product, we show that the existence and uniqueness of fZK equation in \(L^\infty(0,\bar T;H^s_{\mathrm{per}}(Ω))\cap W^{1,\infty}(0,\bar T;L^2_{\mathrm{per}}(Ω))\). We then introduce a modified projection adapted to the fractional transport dispersive operator and prove optimal spatial error estimates of order \(\mathcal O(N^{-r})\) for \(r>2+α\), together with exponential convergence for analytic solutions. An integrating-factor fourth-order four-stage Runge--Kutta time discretization is used to integrate the stiff fractional dispersive part exactly, and a fourth-order temporal error estimate is obtained under a high-regularity nonlinear stability assumption. Numerical experiments illustrate the accuracy, fractional-order dependence, and fully discrete conservation drift of the method.