A “converse” stability condition is necessary for a compact higher order scheme on non-uniform meshes for the time-dependent Schrödinger equation

The stability bounds and error estimates for a compact higher order Numerov-Crank-Nicolson scheme on non-uniform spatial meshes for the 1D time-dependent Schrödinger equation have been recently derived. This analysis has been done in L2 and H1 mesh norms and used the non-standard “converse” condition hω≤c0τ, where hω is the mean spatial step, τ is the time step and c0>0. Now we prove that such condition is necessary for some families of non-uniform meshes and any spatial norm. Also computational results for zero and non-zero potentials show unacceptably wrong behavior of numerical solutions when τ decreases and this condition is violated.