Energy control of the numerical solutions of an elastic oscillator

This paper points out how the energy may be used to select appropriate numerical schemes. For the purpose of illustration an elastic oscillator is considered. This oscillator is modeled by a system of differential equations (zero-dimensional oscillator) and by a system of partial differential equations (one-dimensional oscillator). For the case of free oscillator the energy is preserved in time by the exact solutions. It is shown here that the numerical schemes which preserve this property better give more accurate numerical solutions at a given time and for a given time step. The one-dimensional scheme applies equally well when shock waves are involved. The relation between the solutions obtained with the two models is also discussed.