A compact higher-order finite-difference scheme for the wave equation can be strongly non-dissipative on non-uniform meshes

We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme the 1D homogeneous wave equation in case non-uniform spatial meshes. first show that uniform time cannot be valid any norm provided complex eigenvalues appear associated mesh eigenvalue problem. Moreover, we prove then solution grows exponentially making strongly non-dissipative and therefore impractical. Numerical results confirm this conclusion. In addition, some sequences refining meshes, an excessively strong condition between steps space is (even stability) which familiar explicit schemes parabolic case.