A nonlinear discrete model for approximating a conservative multi-fractional Zakharov system: Analysis and computational simulations
A nonlinear discrete model for approximating a conservative multi-fractional Zakharov system: Analysis and computational simulations
复制标题
DOI：
10.1016/j.matcom.2022.05.026
复制
发表时间：
2022-06
期刊：
影响因子：
--
通讯作者：
Romeo Martínez;J. Macías-Díaz;Q. Sheng
中科院分区：
文献类型：
--
作者：
Romeo Martínez;J. Macías-Díaz;Q. Sheng
本文研究了一类具有分数扩散的两个偏微分方程系统。该系统扩展了传统的Zakharov系统，其中未知量为非线性耦合复值和实值函数。在Riesz意义上理解扩散，并在实数的开放有界区域上施加合适的初始边界条件。结果表明，该体系的质量和希格斯自由能是守恒的。并且证明了总能量是耗散的，自由能量和总能量都是非负的。作为能量守恒的一个推论，我们发现系统的解在时间上是有界的。基于系统解的这些性质，我们提出了一个用有限差分方法逼近分数阶Zakharov系统的数值模型。在求解连续系统的数值模型的同时，我们还提供了质量、希格斯自由能和总能量的离散模拟。利用Browder不动点定理，建立了离散模型的溶解度。结果表明，离散总质量和离散自由能守恒，与连续情况一致。通过数值解的有界性，证明了离散能量泛函（包括离散自由能和离散总能量）是离散时间的非负函数。并对该方案的一致性、稳定性和收敛性进行了严格的研究。数值模拟说明了有限差分解过程的一些预期的理论特征。
A system of two partial differential equations with fractional diffusion is considered in this study. The system extends the conventional Zakharov system with unknowns being nonlinearly coupled complex- and real-valued functions. The diffusion is understood in the Riesz sense, and suitable initial-boundary conditions are imposed on an open and bounded domain of the real numbers. It is shown that the mass and Higgs’ free energy of the system are conserved. Moreover, the total energy is proven to be dissipated, and that both the free and the total energy are non-negative. As a corollary from the conservation of energy, we find that the solutions of the system are bounded throughout time. Motivated by these properties on the solutions of the system, we propose a numerical model to approximate the fractional Zakharov system via finite-difference approaches. Along with this numerical model for solving the continuous system, discrete analogues for the mass, the Higgs’ free energy and the total energy are we provided. Furthermore, utilizing Browder’s fixed-point theorem, we establish the solubility of the discrete model. It is shown that the discrete total mass and the discrete free energy are conserved, in agreement with the continuous case. The discrete energy functionals (both the discrete free energy and the discrete total energy) are proven to be non-negative functions of the discrete time thoroughly the boundedness of the numerical solutions. Properties of consistency, stability and convergence of the scheme are also studied rigorously. Numerical simulations illustrate some of the anticipated theoretical features of our finite-difference solution procedure.