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We study necessary conditions for stability of a Numerov-type compacthigher-order finite-difference scheme for the 1D homogeneous wave equation inthe case of non-uniform spatial meshes. We first show that the uniform in timestability cannot be valid in any spatial norm provided that the complexeigenvalues appear in the associated mesh eigenvalue problem. Moreover, weprove that then the solution norm grows exponentially in time making the schemestrongly non-dissipative and therefore impractical. Numerical results confirmthis conclusion. In addition, for some sequences of refining spatial meshes, anexcessively strong condition between steps in time and space is necessary (evenfor the non-uniform in time stability) which is familiar for explicit schemesin the parabolic case.

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