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Stability and convergence of Dufort-Frankel-type difference schemes for a nonlinear Schrödinger-type equation

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Abstract

We consider a first boundary problem for the nonlinear Schrödinger equation

$$\frac{{\partial u}}{{\partial t}} = ia\frac{{\partial ^2 u}}{{\partial x^2 }} + f(u,u*)u.$$

The convergence of a three-layer explicit difference scheme in theC andW 21 norms is proved. The stability of the scheme with respect to the initial data in the same norms is proved. To justify the convergence and stability we use grid analogues of the energy-preservation laws and grid multiplicative inequalities. The relation 2|a|τ/h 2≤ν<1 is assumed for the grid stepsizes.

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Authors
  1. F. Ivanauskas
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  2. M. Radžiūnas
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Additional information

Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 37, No. 3, pp. 334–352, July–September, 1997.

Translated by V. Mackevičius

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Ivanauskas, F., Radžiūnas, M. Stability and convergence of Dufort-Frankel-type difference schemes for a nonlinear Schrödinger-type equation. Lith Math J 37, 249–263 (1997). https://doi.org/10.1007/BF02465356

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  • DOI: https://doi.org/10.1007/BF02465356

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