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

TL;DR

This paper establishes that a specific converse stability condition is essential for the stability of higher order compact schemes on non-uniform meshes for the Schrödinger equation, supported by theoretical proofs and numerical evidence.

Contribution

It proves the necessity of the converse stability condition for compact higher order schemes on non-uniform meshes, extending previous stability bounds.

Findings

The converse condition is necessary for stability in all space norms.

Violating the condition leads to significant mass non-conservation.

Numerical experiments confirm theoretical predictions.

Abstract

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 Schr\"odinger 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 $h_\omega\leq c_0\tau$, where $h_\omega$ is the mean space 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 space norm. Also numerical results show unacceptably wrong behavior of numerical solutions (their dramatic mass non-conservation) when this condition is violated.