On the Numerical Treatment of an Abstract Nonlinear System of Coupled Hyperbolic Equations Associated with the Timoshenko Model

TL;DR

This paper develops a second-order accurate, parallelizable time-stepping scheme for nonlinear coupled hyperbolic equations related to the Timoshenko model, combining spatial spectral methods and rigorous convergence analysis.

Contribution

It introduces a symmetric three-layer semi-discrete scheme that simplifies nonlinear problems to linear ones at each step, with proven convergence and efficiency in spectral spatial discretization.

Findings

The scheme achieves second-order accuracy in time.

Numerical experiments confirm the theoretical convergence.

Efficient decoupling of linear systems via Legendre-Galerkin spectral method.

Abstract

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 then 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.