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

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.

Keywords

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

2010 Mathematics Subject Classification

Primary 81Q05. Secondary 65M06, 65M12.

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Published 1 January 2006