Remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis

Russian Journal on Numerical Analysis and Mathematical Modelling (Germany). 2016. Vol. 31. No. 1. P. 51–64.

Zlotnik Alexander, Zlotnik Ilya

We consider the generalized time-dependent Schrödinger equation on the half-axis and a broad family of finite-difference schemes with the discrete transparent boundary conditions (TBCs) to solve it. We first rewrite the discrete TBCs in a simplified form explicit in space step $h$. Next, for a selected scheme of the family, we discover that the discrete convolution in time in the discrete TBC does not depend on $h$ and, moreover, it coincides with the corresponding convolution in the semi-discrete TBC rewritten similarly. Both moments allow us to prove the bound for the difference between the kernels of the discrete convolutions in the discrete and semi-discrete TBCs that is the first result of such kind. Numerical experiments on replacing the discrete TBC convolutions by the semi-discrete one exhibit truly small absolute errors though not relative ones in general. The suitable discretization in space of the semi-discrete TBC for the higher-order Numerov scheme is also discussed.

Keywords: The time-dependent Schrödinger equation, approximate and discrete transparent boundary conditions, discrete convolution operator, приближенные и дискретные прозрачные граничные условия, finite-difference schemes, разностные схемы, нестационарное уравнение Шрёдингера, the Numerov discretization in space, дискретизация Нумерова по пространству, дискретная свертка

Publication based on the results of: Построение и анализ численных методов решения некоторых задач для эволюционных уравнений математической физики (2014)