关于有限元法在时间上的里德伯格外推用于一维薛定谔方程初值问题的离散透明边界条件
On the Richardson Extrapolation in Time of Finite Element Method with Discrete TBCs for the Cauchy Problem for the 1D Schrödinger Equation
AI总结：
本文提出利用里德伯格外推提高有限元法在时间上的精度，通过数值实验展示了其在解决一维薛定谔方程初值问题时的高精度和低计算成本优势。
摘要：
我们考虑了一维广义薛定谔方程在整个轴上的初值问题。为解决这个问题，最近已构建了任意阶空间有限元和时间上的Crank-Nicolson方法，并结合离散透明边界条件（TBCs）。现在我们利用里德伯格外推来显著提高时间步长的精度。为了研究其性质，我们给出了三个典型例子的数值实验结果和扩大了的实际误差分析。所得到的方法能够在合理的计算成本下提供高精度结果，这在空间或时间步长的更常用二次阶方法中是无法达到的。与之前的结果相比，我们使用更少的元素和时间步长获得了更精确的结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
