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.

在这项工作中，我们提出了一种隐式有限差分格式来逼近著名的 Klein-Gordon-Zakharov 系统泛化的解。更准确地说，这项工作中考虑的系统是对经典 Klein-Gordon-Zakharov 模型空间分数情况的扩展，考虑了两种不同的微分阶和 Riesz 类型的分数导数。在这项工作中提出的数值模型考虑分数阶中心差异来近似空间分数阶导数。与这个离散系统相关的能量是一个非负不变量，与连续分数模型的属性一致。我们使用定点参数和复矩阵属性严格建立解的存在性。为此，我们使用离散化的两个差分方程解耦的事实，这意味着计算实现比文献中可用的其他数值模型更容易。我们证明了该方法在时间和空间上都具有平方一致性。此外，我们严格证明了数值模型的稳定性和二次收敛性。作为稳定性的必然结果，我们能够证明数值解的唯一性。最后，我们提供了一些说明性的模拟与我们的方案的计算机实现。我们严格证明了数值模型的稳定性和二次收敛性。作为稳定性的必然结果，我们能够证明数值解的唯一性。最后，我们提供了一些说明性的模拟与我们的方案的计算机实现。我们严格证明了数值模型的稳定性和二次收敛性。作为稳定性的必然结果，我们能够证明数值解的唯一性。最后，我们提供了一些说明性的模拟与我们的方案的计算机实现。