Abstract In this work, we investigate a multidimensional system consisting of a finite (though arbitrary) number of coupled hyperbolic partial differential equations with fractional diffusion, constant damping and inertial times, and nonlinear reaction terms. Under suitable analytical conditions, the model has conserved quantities which are preserved in the absence of damping. We establish rigorously the conservation of the proposed quantities and, assuming that solutions of the model exist, we prove their boundedness. Motivated by these facts, we propose a finite-difference methodology to approximate the solutions of the continuous system. As its continuous counterpart, the discrete model has associated discrete quantities that estimate the Hamiltonian functional. Moreover, these quantities are preserved in the absence of damping, and they are dissipated when damping is present. To prove this feature of our finite-difference scheme, a new approximation form of the nonlinear reaction terms is proposed. This approach allows for the scheme to mimic the properties of the continuous system. The numerical properties of consistency, stability, boundedness and convergence of the scheme are proved rigorously. Some illustrative simulations confirm that the scheme is capable of preserving or dissipating the quantities, in agreement with the analytical results.

中文翻译：

---

耦合非线性分数双曲方程多维系统的全显式变分积分器

摘要 在这项工作中，我们研究了一个多维系统，该系统由有限（尽管任意）数量的耦合双曲偏微分方程组成，这些方程具有分数扩散、恒定阻尼和惯性时间以及非线性反应项。在合适的分析条件下，模型具有在没有阻尼的情况下保持不变的守恒量。我们严格地建立了建议数量的守恒，并且假设模型的解存在，我们证明了它们的有界性。受这些事实的启发，我们提出了一种有限差分方法来逼近连续系统的解。作为其连续对应物，离散模型具有相关的离散量，用于估计哈密顿函数。此外，这些量在没有阻尼的情况下保持不变，当存在阻尼时，它们就会消散。为了证明我们的有限差分格式的这一特征，提出了非线性反应项的新近似形式。这种方法允许该方案模拟连续系统的特性。严格证明了该方案的一致性、稳定性、有界性和收敛性的数值性质。一些说明性的模拟证实，该方案能够保持或消散数量，与分析结果一致。严格证明了该方案的有界性和收敛性。一些说明性的模拟证实，该方案能够保持或消散数量，与分析结果一致。严格证明了该方案的有界性和收敛性。一些说明性的模拟证实，该方案能够保持或消散数量，与分析结果一致。