关于与Timoshenko模型相关的抽象非线性耦合双曲方程组数值处理
On the Numerical Treatment of an Abstract Nonlinear System of Coupled Hyperbolic Equations Associated with the Timoshenko Model
AI总结：
本文研究了与Timoshenko模型相关的抽象非线性耦合双曲方程组的Cauchy问题，提出了一种对称的三层半离散时间步进方案，通过时间中点评估非线性项，将原问题转化为线性问题并实现并行计算，证明了方法的收敛性及时间步长的二阶精度。
摘要：
本文研究了与Timoshenko模型相关的抽象非线性耦合双曲方程组的Cauchy问题。目的是开发并深入探讨一种时间离散化方案，以近似该问题的解。为此，我们提出了一种对称的三层半离散时间步进方案，其中非线性项在时间中点处进行评估。因此，在每个时间步中，该方法将原始非线性问题转化为线性问题，并能够并行计算其解。证明了收敛性，并在局部时间区间内建立了关于时间步长的二阶精度。所提出的方案应用于一维非线性动态Timoshenko梁系统，并将抽象非线性系统的结果扩展到此设置。采用Legendre-Galerkin谱近似进行空间离散化。通过在Galerkin框架内取Legendre多项式之差，所得到的线性系统是稀疏的，可以高效解耦。还研究了该方法的收敛性。最后，进行了若干在精心选择的基准问题上的数值实验，以验证所提出的方法并确认理论发现。
英文摘要：
The present work addresses the Cauchy problem for an abstract nonlinear system of coupled hyperbolic equations associated with the Timoshenko model in a real Hilbert space. Our purpose is to develop and delve into a temporal discretization scheme for approximating a solution to this problem. To this end, we propose a symmetric three-layer semi-discrete time-stepping scheme in which the nonlinear term is evaluated at the temporal midpoint. As a result, at each time step, this approach reduces the original nonlinear problem to a linear one and enables parallel computation of its solution. Convergence is proved, and second-order accuracy with respect to the time-step size is established on a local temporal interval. The proposed scheme is applied to a spatially one-dimensional nonlinear dynamic Timoshenko beam system, and the results obtained for the abstract nonlinear system are extended to this setting. A Legendre-Galerkin spectral approximation is employed for the spatial discretization. By taking differences of Legendre polynomials within the Galerkin framework, the resulting linear system is sparse and can be efficiently decoupled. The convergence of the method is also investigated. Finally, several numerical experiments on carefully chosen benchmark problems are conducted to validate the proposed approach and to confirm the theoretical findings.
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