Volume 14, Issue 6
Two-Grid Crank-Nicolson Finite Volume Element Method for the Time-Dependent Schrödinger Equation

Adv. Appl. Math. Mech., 14 (2022), pp. 1357-1380.

Published online: 2022-08

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• Abstract

In this paper, we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schrödinger equation. Combining the idea of two-grid discretization, the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space, which can ensure the accuracy while the size of coarse grid is much coarser than that of fine grid. We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator. Finally, numerical simulations are provided to verify the correctness of the theoretical analysis.

• Keywords

Finite volume element method, two-grid method, Crank-Nicolson scheme, error estimates, Schrödinger equation.

• AMS Subject Headings

65N12, 65M60

• BibTex
• RIS
• TXT
@Article{AAMM-14-1357, author = {Chuanjun and Chen and and 24232 and and Chuanjun Chen and Yuzhi and Lou and and 24233 and and Yuzhi Lou and Tong and Zhang and and 24234 and and Tong Zhang}, title = {Two-Grid Crank-Nicolson Finite Volume Element Method for the Time-Dependent Schrödinger Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {6}, pages = {1357--1380}, abstract = {

In this paper, we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schrödinger equation. Combining the idea of two-grid discretization, the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space, which can ensure the accuracy while the size of coarse grid is much coarser than that of fine grid. We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator. Finally, numerical simulations are provided to verify the correctness of the theoretical analysis.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0233}, url = {http://global-sci.org/intro/article_detail/aamm/20851.html} }
TY - JOUR T1 - Two-Grid Crank-Nicolson Finite Volume Element Method for the Time-Dependent Schrödinger Equation AU - Chen , Chuanjun AU - Lou , Yuzhi AU - Zhang , Tong JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1357 EP - 1380 PY - 2022 DA - 2022/08 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0233 UR - https://global-sci.org/intro/article_detail/aamm/20851.html KW - Finite volume element method, two-grid method, Crank-Nicolson scheme, error estimates, Schrödinger equation. AB -

In this paper, we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schrödinger equation. Combining the idea of two-grid discretization, the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space, which can ensure the accuracy while the size of coarse grid is much coarser than that of fine grid. We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator. Finally, numerical simulations are provided to verify the correctness of the theoretical analysis.

Chuanjun Chen, Yuzhi Lou & Tong Zhang. (2022). Two-Grid Crank-Nicolson Finite Volume Element Method for the Time-Dependent Schrödinger Equation. Advances in Applied Mathematics and Mechanics. 14 (6). 1357-1380. doi:10.4208/aamm.OA-2021-0233
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