A converse stability condition is necessary for a compact higher order scheme on non-uniform meshes for the time-dependent Schrodinger 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 Schrodinger 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(omega) <= c(0)tau where h(omega), is the mean spatial step, tau 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 spatial norm. Also computational results for zero and non-zero potentials show unacceptably wrong behavior of numerical solutions when tau decreases and this condition is violated. (C) 2018 Elsevier Ltd. All rights reserved.