A Programming Code for the Numerical Treatment of an Abstract Nonlinear System of Coupled Hyperbolic Equations Associated with the Timoshenko Model

This source code corresponds to the combined numerical schemes developed to obtain numerical solutions of an abstract nonlinear system of coupled hyperbolic equations arising from the Timoshenko model. We propose a symmetric semi-discrete time-stepping scheme for this system, which reduces the original problem to a linear one and allows for parallel computation of its solution. Subsequently, a one-dimensional nonlinear dynamic Timoshenko beam system is considered, and the Legendre–Galerkin spectral method is employed for spatial discretization. Choosing differences of Legendre polynomials in the Galerkin method yields a sparse linear system that can be decoupled. The inner products used in the proposed algorithm are computed approximately using Gauss–Legendre quadrature.