On the propagation of round-off errors in the numerical treatment of the wave equation

Abstract:

An upper bound of the norm of the error vector after n time steps is $\tfrac {1}{2}(n + 1)(n + 2)\| {{\delta ^{\ast }}}\|$. For the explicit scheme ${\delta ^{\ast }} = \| {{\delta ^{\ast }}}\| = 3 \times \tfrac {1}{2} \times {10^{ - p}}$ where p is the number of decimals carried in the computations. For the implicit scheme ${\delta ^{\ast }} = \| {{\delta ^{\ast }}}\|$ is an upper bound of the errors which arise both from using approximations to ${A^{ - 1}}$ and ${A^{ - 1}}B$ in the determination of ${u_{k + 1}}$ from equation (6$^{*}$) and from rounding off the values of the products and quotients involved in the computation of the components of ${u_{k + 1}}$.