The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis

Alexander Zlotnik

doi:10.3934/krm.2015.8.587

Article Contents

2015, Volume 8, Issue 3: 587-613. Doi: 10.3934/krm.2015.8.587

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The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis

Alexander Zlotnik^1,

1.
Department of Mathematics at Faculty of Economics Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow

Received: October 31, 2014

Revised: February 28, 2015

Published: August 2015

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Abstract

We deal with the initial-boundary value problem for the 1D time-dependent Schrödinger equation on the half-axis. The finite-difference scheme with the Numerov averages on the non-uniform space mesh and of the Crank-Nicolson type in time is studied, with some approximate transparent boundary conditions (TBCs). Deriving bounds for the skew-Hermitian parts of the Numerov sesquilinear forms, we prove the uniform in time stability in $L^2$- and $H^1$-like space norms under suitable conditions on the potential and the meshes. In the case of the discrete TBC, we also derive higher order in space error estimates in both norms in dependence with the Sobolev regularity of the initial function (and the potential) and properties of the space mesh. Numerical results are presented for tunneling through smooth and rectangular potentials-wells, including the global Richardson extrapolation in time to ensure higher order in time as well.

Keywords:

Time-dependent Schrödinger equation,
unbounded domain,
Numerov scheme,
Crank-Nicolson scheme,
approximate transparent boundary conditions,
stability,
error estimates,
global Richardson extrapolation.

Mathematics Subject Classification: Primary: 65M06, 65M12, 65M15; Secondary: 35Q41.

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