Discrete energy conservation law and reciprocity relationship for one-dimensional wave propagation problems

A mathematical derivation is presentcd to establish the discrete energy conservation law and reciprocity relationship for explaining the insensitive phenomenon round in the previous investigations of a one-dimensional wave propagation problem. It is shown that there do exist certain discretized differential and variational systems which can preserve the energy conservation and reciprocity relationships. This then excludes the use of these relationships in error estimation.