The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrὅdinger Equation

Computational Methods in Applied Mathematics. 2015. Vol. 15. No. 2. P. 233–245.

Zlotnik Alexander, Zlotnik Ilya

We consider the Cauchy problem for the 1D generalized Schrὅdinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transpa\-rent boundary conditions (TBCs) has recently been constructed. Now we engage the global Richardson extrapolation in time to derive the high order method both with respect to space and time steps. To study its properties, we comment on its stability and give results of numerical experiments and enlarged practical error analysis for three typical examples. Unlike most common 2nd order (in either space or time step) methods, the proposed method is able to provide high precision results in the uniform norm by using adequate computational costs. It works even in the case of discontinuous potentials and non-smooth solutions far beyond the scope of its standard theory. Comparing our results to the previous ones, we obtain \textit{much more} accurate results using much less amount of both elements and time steps.

Keywords: метод конечных элементов, задача Коши, the Cauchy problem, устойчивость, discrete transparent boundary conditions, дискретные прозрачные граничные условия, The time-dependent Schrödinger equation, finite element method, нестационарное уравнение Шрёдингера, practical error analysis, практический анализ погрешности, stability, the Richardson extrapolation, экстраполяция Ричардсона

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