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.

中文翻译：

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一种双分数保守Klein-Gordon-Zakharov系统的节能高效方案

摘要 在这项工作中，我们考虑了 Klein-Gordon-Zakharov 系统的分数扩展，该系统描述了在高频等离子体中朗缪尔波上强湍流的传播。两个组件都考虑了 Riesz 类型的空间分数导数，并且初始边界条件被强加在实数的闭合和有界区间上。在第一阶段，我们证明系统的总能量是守恒的，并且系统的全局解是有界的。受这些结果的启发，我们提出了一种有限差分方案来逼近解，以及能量泛函的离散形式。这项工作中提出的离散化的优点在于求解分量方程的差分方程是解耦的。这意味着可以在每个时间步骤单独求解数值方案。我们严格地建立了解的存在性，以及方案保存离散能量的能力。该方法在空间和时间上都具有二阶一致性。此外，使用能量方法的离散形式，我们在数学上建立了有限差分格式是稳定的和二次收敛的。我们提供了一些模拟来表明所提出的方法是二次收敛的，并且它保留了系统的总能量。我们在数学上建立了有限差分格式是稳定的和二次收敛的。我们提供了一些模拟来表明所提出的方法是二次收敛的，并且它保留了系统的总能量。我们在数学上建立了有限差分格式是稳定的和二次收敛的。我们提供了一些模拟来表明所提出的方法是二次收敛的，并且它保留了系统的总能量。