On the Richardson extrapolation in time of finite element method with discrete TBCs for the Cauchy problem for the 1D Schrödinger equation

2014. No. arXiv:1405.3147.

Zlotnik Alexander, Zlotnik Ilya

We consider the Cauchy problem for the 1D generalized Schrödinger 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 Richardson extrapolation to improve significantly the accuracy in time step. To study its properties, we give results of numerical experiments and enlarged practical error analysis for three typical examples. The resulting method is able to provide high precision results in the uniform norm for reasonable computational costs that is unreachable by more common 2nd order methods in either space or time step. Comparing our results to the previous ones, we obtain much more accurate results using much less amount of both elements and time steps.

Keywords: задача Коши, the Cauchy problem, устойчивость, discrete transparent boundary conditions, дискретные прозрачные граничные условия, The time-dependent Schrödinger equation, finite element method, метод конечных элементов (МКЭ), 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)