On the Richardson Extrapolation in Time of Finite Element Method with Discrete TBCs for the Cauchy Problem for the 1D Schrödinger Equation

TL;DR

This work addresses the Cauchy problem for the $1$D Schrödinger equation on the real line by coupling a high-order finite element discretization in space with Crank–Nicolson time stepping and discrete transparent boundary conditions. The authors introduce Richardson extrapolation in time to boost temporal accuracy while preserving stability and the discrete TBC framework, supported by a rigorous energy-stability analysis and an explicit time-error expansion. Numerical experiments on Gaussian propagation, tunneling through a barrier, and a double-barrier quantum well demonstrate that high-order Richardson extrapolation (orders $r=2,3,4$) yields substantial accuracy gains at modest additional cost, often achieving uniform-norm precision far beyond standard second-order schemes. The results show how, with suitable mesh parameters and extrapolation order, one can obtain highly accurate solutions with far fewer spatial elements and time steps than previously reported, highlighting practical benefits for quantum-mechanical simulations in unbounded domains.

Abstract

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 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.

Figures (18)

• Figure 1: Example 1. $|\Psi_{4R}^{m}|$ and $\mathop{\mathrm{Re}}\nolimits\Psi_{4R}^{m}$ for $n=9$ and $(J,M)=(30,300)$
• Figure 2: Example 1. The errors $\max_{0\leqslant rm\leqslant M}\|\psi^{rm}-\Psi_{rR}^{rm}\|$, for $n=9$ and $J=90$, in dependence with $r=1,2,3,4$ and $M=300,600,\dots,3000$
• Figure 3: Example 1. The errors: (a) $\|\psi^{3m}-\Psi^{(k\tau),\,3m}\|$ for $k=1,3/2,3$ and $\|\psi^{3m}-\Psi_{3R}^{3m}\|$ ($r=3$), and (b) $\|\psi^{4m}-\Psi^{(k\tau),\,4m}\|$ for $k=1,4/3,2,4$ and $\|\psi^{4m}-\Psi_{4R}^{4m}\|$ ($r=4$), both for $n=9$ and $(J,M)=(90,1500)$, in dependence with $t_m$
• Figure 4: Example 1. The errors $\max_{0\leqslant 3m\leqslant M}\|\psi^{3m}-\Psi_{3R}^{3m}\|_{C_h}$, for $M=3000$, in dependence with $n$ and $J$
• Figure 5: Example 2. $|\Psi_{4R}^{m}|$ and $\mathop{\mathrm{Re}}\nolimits\Psi_{4R}^{m}$, for $n=9$ and $(J,M)=(60,576)$, and the scaled $V$

Theorems & Definitions (2)

• Proposition 1
• Proposition 2