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

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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. Bibliography: 12 titles.

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Authors and Affiliations

  1. Russian State Social University, 4, V. Pika ul., Moscow, 129226, Russia

    A. A. Zlotnik

  2. Moscow Power Engineering Institute (Technical University), 14, Krasnokazarmennaya ul., Moscow, 111250, Russia

    A. A. Zlotnik & A. V. Lapukhina

Authors
  1. A. A. Zlotnik
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  2. A. V. Lapukhina
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Correspondence to A. A. Zlotnik.

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Translated from Problems in Mathematical Analysis 47, June 2010, pp. 77–88

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Zlotnik, A.A., Lapukhina, A.V. Stability of a Numerov type finite–difference scheme with approximate transparent boundary conditions for the nonstationary Schrödinger equation on the half-axis. J Math Sci 169, 84–97 (2010). https://doi.org/10.1007/s10958-010-0040-9

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