On the Richardson Extrapolation in Time of Finite Element Method with Discrete TBCs for the Cauchy Problem for the 1D Schr\"odinger Equation

TL;DR

This paper enhances the accuracy of finite element methods for the 1D Schrödinger equation by applying Richardson extrapolation to discrete transparent boundary conditions, achieving high precision with reduced computational effort.

Contribution

It introduces the use of Richardson extrapolation with finite element and Crank-Nicolson methods for the 1D Schrödinger equation, significantly improving temporal accuracy and efficiency.

Findings

High-precision results in the uniform norm achieved

Significant accuracy improvement over standard methods

Reduced computational costs with fewer elements and time steps

Abstract

We consider the Cauchy problem for the 1D generalized Schr\"odinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transparent 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.