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

2017/07/31 by Zlotnik, Alexander, Čiegis, Raimondas
#65M06 #65M12 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1707.09943

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 L² and H¹ mesh norms and used the non-standard "converse" condition h_ω≤ c₀τ, where h_ω is the mean space step, τ is the time step and c₀>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.

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