On long time error bounds for the wave equation on second order form (Q1785490)

The classical convergence for a stable and consistent numerical scheme does not mean that the error is bounded when time increases indefinitely. Consequently, this classical definition is not sufficient for an accurate solution after long time integration. Long time error bounds have been previously studied for hyperbolic problems of first-order form. In this note, the authors show that, with appropriate choices of boundary conditions, the space and time derivatives of the errors are bounded, even for long times. No long time bound on the error itself is obtained in spite of the fact that the numerical experiments indicate that such a bound exists.