Instability of Finite Difference Schemes Due to Boundary Conditions in Elastic Media*

A bstract The manner in which boundary conditions are approximated and introduced into finite difference schemes may have an important influence on the stability and accuracy of the results. The standard von Neumann condition for stability applies only for points which are not in the vicinity of the boundaries. This stability condition does not take into consideration the effects caused by introducing the boundary conditions to the scheme. Working on elastic media with free stress boundary conditions we found that the boundary approximation gives rise to serious stability problems especially for regions with high Poisson's ratio. In order to detect these effects apriori and to analyse them, we have used a more elaborate procedure for checking the stability of the scheme which takes into consideration the boundary conditions. It is based on studying a locally spaced time propagating matrix which governs the time‐space behavior of a small region of the grid which includes free surface points. By using this procedure a better insight into the nature of instability caused by the approximations to the boundary conditions was gained which led us to a new stable approximation for the free surface boundary conditions.