As the first step towards a general analysis of the stability of optimally accurate predictor–corrector (P–C) time domain discretized schemes for solving the elastic equation of motion, we analyze the stability of two P–C schemes for a 1-D homogeneous case. Letting Δt be the time step, h be the spatial grid interval, β be the velocity of seismic wave propagation and be the dimensionless Courant parameter, we find that each scheme has the following stability properties: stability for , instability for , stability for and instability for , where . We refer to the region as the second island of stability. The values of , and are scheme-dependent. The existence of a second island of stability in a numerical scheme for solving the wave equation has not, to our knowledge, been previously reported.