Stability of a Numerov type finite–difference scheme with approximate transparent boundary conditions for the nonstationary Schrödinger equation on the half-axis

Journal of Mathematical Sciences. 2010. Vol. 169. No. 1. P. 84–97.

Злотник А. А., Lapukhina A. V.

We consider an initial-boundary value problem for the one-dimensional nonstationary Schrödinger equation on the half-axis and study a two-level symmetric finite-difference scheme of Numerov type with higher approximation order. This scheme is constructed on a finite mesh, which is uniform with respect to space, with a nonlocal approximate transparent boundary condition of a general form (of Dirichlet-to-Neumann type). We obtain assertions about the stability of the finite-difference scheme in two norms with respect to the initial data and free terms in the equation and in the approximate transparent boundary condition under suitable conditions in the form of inequalities on the operator of approximate transparent boundary condition.

Ключевые слова: устойчивость, The time-dependent Schrödinger equation, approximate and discrete transparent boundary conditions, приближенные и дискретные прозрачные граничные условия, unbounded domains, неограниченные области, нестационарное уравнение Шрёдингера, the Numerov discretization in space, дискретизация Нумерова по пространству, stability