An implicit semi-linear discretization of a bi-fractional Klein–Gordon–Zakharov system which conserves the total energy

Abstract

In this work, we propose an implicit finite-difference scheme to approximate the solutions of a generalization of the well-known Klein–Gordon–Zakharov system. More precisely, the system considered in this work is an extension to the spatially fractional case of the classical Klein–Gordon–Zakharov model, considering two different orders of differentiation and fractional derivatives of the Riesz type. The numerical model proposed in this work considers fractional-order centered differences to approximate the spatial fractional derivatives. The energy associated to this discrete system is a non-negative invariant, in agreement with the properties of the continuous fractional model. We establish rigorously the existence of solutions using fixed-point arguments and complex matrix properties. To that end, we use the fact that the two difference equations of the discretization are decoupled, which means that the computational implementation is easier than for other numerical models available in the literature. We prove that the method has square consistency in both time and space. In addition, we prove rigorously the stability and the quadratic convergence of the numerical model. As a corollary of stability, we are able to prove the uniqueness of numerical solutions. Finally, we provide some illustrative simulations with a computer implementation of our scheme.