On a fractional generalization of a nonlinear model in plasma physics and its numerical resolution via a multi-conservative and efficient scheme

Abstract

In this work, we extend the Zakharov–Rubenchik system to the fractional case by using Riesz operators of fractional order in space. We prove that the system is capable of preserving extensions of the mass, energy, momentum and two linear functionals. In a second stage, we propose a discretization to approximate the solutions of our model. In the way, we propose discrete forms of the conserved functionals, and we prove that they are also conserved in the discrete domain. We prove that the numerical scheme has second-order accuracy in both space and time. Moreover, we establish theoretically the properties of conditional stability and second-order convergence of the scheme. The numerical model was implemented computationally, and some simulations are provided in order to illustrate that the method is capable of conserving the discrete functionals and its rate of convergence. This is the first report in the literature in which a multi-conservative fractional extension of this system is proposed, and a numerical scheme to approximate its solutions is designed and fully analyzed for conservative and numerical properties. Even in the integer-order case, this is the first article which proves the approximate conservation of the momentum, and which rigorously proves the stability and the convergence of a scheme for the Zakharov–Rubenchik system.