Numerical methods for a class of nonlinear hyperbolic problems: a comparative study

In this paper we consider the problem of the free longitudinal vibrations of a beam made of a bimodular material, i.e. an elastic material whose in-tension Young's modulus is a fraction of that under compression. After recalling the exact solutions for an ifinite beam and for a beam with fixed ends calculated via the characteristics method, we apply high-resolution methods based on the finite-element approach to solve the nonliner equation of the motion. In particular, we compare the exact solutions with the numerical solutions calculated using the space-time element method, the collocation and least-squares method, as well as TVD and Newmark methods.