The governing system of equations of the dynamic one-dimensional linear thermoelas ticity is a composed hyperbolic-parabolic system and has been studied little in the literature. A Godunov-type numerical scheme for initial-boundary value problems is constructed. An energy inequality is used to control the stability of the proposed numerical scheme, which also allows us to classify thermoelastic problems in strongly coupled and weakly coupled problems. Most real materials fall into the weakly coupled case when the mechanical power can be neglected in the balance of energy equation. The proposed numerical scheme is used to test that the above property of the weakly coupled case and also to see if, in the strongly coupled case, the solutions of certain initial-boundary value problems show oscillatory behavior, which has certain similarities to those experimentally recorded at very low temperature and thought of as evidence of the existence of the "second sound."