A numerical method for longitudinal wave propagation in nonlinear elastic solids is presented. Here, we consider polynomial stress-strain relationships, which are widely used in nondestructive evaluation. The large-strain and infinitesimal-strain constitutive laws deduced from Murnaghan'sl aw are detailed, and polynomial expressions are obtained. The Lagrangian equations of motion yield ahyperbolic system of conservation laws. The latter is solved numerically using afi nite-volume method with flux limiters based on Roe linearization. The method is tested on the Riemann problem, which yields nonsmooth solutions. The method is then applied to acontinuum model with one scalar internal variable, accounting for the softening of the material (slowdynamics).