Communications in Mathematical Sciences

Volume 4 (2006)

Number 4

On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I

Pages: 741 – 766



B. Ducomet

B. Ducomet

A. Zlotnik

A. Zlotnik


We consider initial-boundary value problems for a generalized time-dependent Schrödinger equation in $1D$ on the semi-axis and in $2D$ on a semi-bounded strip. For Crank-Nicolson finite-difference schemes, we suggest an alternative coupling to approximate transparent boundary conditions and present a condition ensuring unconditional stability. In the case of discrete transparent boundary conditions, we revisit the statement and the proof of stability together with the derivation of the conditions.


Stability; Crank-Nicolson; finite-difference scheme; transparent boundary conditions; time-dependent Schrödinger equation

2010 Mathematics Subject Classification

Primary 81Q05. Secondary 65M06, 65M12.

Published 1 January 2006