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

Applied Mathematics Letters. 2018. Vol. 80. P. 35–40.

Zlotnik A., Čiegis R.

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 $h_\omega\leq c_0\tau$, where $h_\omega$ is the mean spatial step, $\tau$ is the time step and $c_0>0$. Now we prove that such condition is \textit{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.

Keywords: устойчивость, The time-dependent Schrödinger equation, нестационарное уравнение Шрёдингера, stability, Numerov-Crank-Nicolson scheme, схема Нумерова-Кранка-Никольсон, non-uniform spatial mesh, неравномерная сетка по пространству

Publication based on the results of: Анализ погрешности численных методов решения задач математической физики на классах данных (2016)