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

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 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 $h_{\omega }\le c_0\tau$, 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.

6 pages, 2 figures, 1 table

References in corpus (2)

• A direct fast FFT-based implementation for high order finite element method on rectangular parallelepipeds for PDE
• A proposal for sympathetically cooling neutral molecules using cold ions

Cited by in corpus (2)

• On compact 4th order finite-difference schemes for the wave equation
• A compact higher-order finite-difference scheme for the wave equation can be strongly non-dissipative on non-uniform meshes