A "converse" stability condition is necessary for a compact higher order scheme on non-uniform meshes

The stability bounds and error estimates for a compact higher order Numerov-Crank-Nicolson scheme on non-uniform space meshes for the 1D time-dependent Schrödinger equation have been recently derived. This analysis has been done in $L^2$ and $H^1$ mesh norms and used the non-standard "converse" condition $h_ω\leq c_0τ$, where $h_ω$ is the mean space step, $τ$ is the time step and $c_0>0$. Now we prove that such condition is necessary for some families of non-uniform meshes and any space norm. Also numerical results show unacceptably wrong behavior of numerical solutions (their dramatic mass non-conservation) when this condition is violated.

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