A "converse" stability condition is necessary for a compact higher order scheme on non-uniform meshes for the time-dependent Schrödinger equation. (June 2018) - Zlotnik, Alexander; Čiegis, Raimondas
Abstract: 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 L 2 and H 1 mesh norms and used the non-standard "converse" condition h ω ≤ c 0 τ, where h ω is the mean spatial 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 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.