An energy-preserving and efficient scheme for a double-fractional conservative Klein–Gordon–Zakharov system

Abstract

In this work, we consider a fractional extension of the Klein–Gordon–Zakharov system which describes the propagation of strong turbulences on the Langmuir wave in a high-frequency plasma. Both components consider space-fractional derivatives of the Riesz type, and initial-boundary conditions are imposed on a closed and bounded interval of the real numbers. In a first stage, we show that the total energy of the system is conserved, and that the global solutions of the system are bounded. Motivated by these results, we propose a finite-difference scheme to approximate the solutions, and a discrete form of the energy functional. The advantage of the discretization proposed in this work lies in that the difference equations to solve the component equations are decoupled. This implies that the numerical schemes can be solved separately at each temporal step. We establish rigorously the existence of solutions, as well as the capability of the scheme to conserve the discrete energy. The method has a second-order consistency in both space and time. Moreover, using a discrete form of the energy method, we establish mathematically that the finite-difference scheme is stable and quadratically convergent. We provide some simulations to show that the proposed methodology is quadratically convergent and that it preserves the total energy of the system.