An alternating direction implicit finite element Galerkin method for the linear Schrödinger equation

Morrakot Khebchareon; Amiya K. Pani; Graeme Fairweather; Ryan I. Fernandes

doi:10.1007/s11075-023-01740-5

An alternating direction implicit finite element Galerkin method for the linear Schrödinger equation

Morrakot Khebchareon
, Amiya K. Pani
, Graeme Fairweather
, Ryan I. Fernandes

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We formulate and analyze a fully discrete approximate solution of the linear Schrödinger equation on the unit square written as a Schrödinger-type system. The finite element Galerkin method is used for the spatial discretization, and the time stepping is done with an alternating direction implicit extrapolated Crank-Nicolson method. We demonstrate the existence and uniqueness of the approximation, and prove that the scheme is of optimal accuracy in the L², H¹ and L^∞ norms in space and second-order accurate in time. Numerical results are presented which support the theory.

Original language	British English
Journal	Numerical Algorithms
DOIs	https://doi.org/10.1007/s11075-023-01740-5
State	Accepted/In press - 2024

Keywords

Alternating direction implicit method
Extrapolated Crank-Nicolson method
Finite element Galerkin method
Linear Schrödinger equation
Numerical experiments
Optimal-order convergence
Schrödinger-type systems

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10.1007/s11075-023-01740-5

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title = "An alternating direction implicit finite element Galerkin method for the linear Schr{\"o}dinger equation",

abstract = "We formulate and analyze a fully discrete approximate solution of the linear Schr{\"o}dinger equation on the unit square written as a Schr{\"o}dinger-type system. The finite element Galerkin method is used for the spatial discretization, and the time stepping is done with an alternating direction implicit extrapolated Crank-Nicolson method. We demonstrate the existence and uniqueness of the approximation, and prove that the scheme is of optimal accuracy in the L2, H1 and L∞ norms in space and second-order accurate in time. Numerical results are presented which support the theory.",

keywords = "Alternating direction implicit method, Extrapolated Crank-Nicolson method, Finite element Galerkin method, Linear Schr{\"o}dinger equation, Numerical experiments, Optimal-order convergence, Schr{\"o}dinger-type systems",

author = "Morrakot Khebchareon and Pani, \{Amiya K.\} and Graeme Fairweather and Fernandes, \{Ryan I.\}",

note = "Publisher Copyright: {\textcopyright} 2024, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.",

year = "2024",

doi = "10.1007/s11075-023-01740-5",

language = "British English",

journal = "Numerical Algorithms",

issn = "1017-1398",

publisher = "Springer Netherlands",

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TY - JOUR

T1 - An alternating direction implicit finite element Galerkin method for the linear Schrödinger equation

AU - Khebchareon, Morrakot

AU - Pani, Amiya K.

AU - Fairweather, Graeme

AU - Fernandes, Ryan I.

PY - 2024

Y1 - 2024

N2 - We formulate and analyze a fully discrete approximate solution of the linear Schrödinger equation on the unit square written as a Schrödinger-type system. The finite element Galerkin method is used for the spatial discretization, and the time stepping is done with an alternating direction implicit extrapolated Crank-Nicolson method. We demonstrate the existence and uniqueness of the approximation, and prove that the scheme is of optimal accuracy in the L2, H1 and L∞ norms in space and second-order accurate in time. Numerical results are presented which support the theory.

AB - We formulate and analyze a fully discrete approximate solution of the linear Schrödinger equation on the unit square written as a Schrödinger-type system. The finite element Galerkin method is used for the spatial discretization, and the time stepping is done with an alternating direction implicit extrapolated Crank-Nicolson method. We demonstrate the existence and uniqueness of the approximation, and prove that the scheme is of optimal accuracy in the L2, H1 and L∞ norms in space and second-order accurate in time. Numerical results are presented which support the theory.

KW - Alternating direction implicit method

KW - Extrapolated Crank-Nicolson method

KW - Finite element Galerkin method

KW - Linear Schrödinger equation

KW - Numerical experiments

KW - Optimal-order convergence

KW - Schrödinger-type systems

UR - https://www.scopus.com/pages/publications/85182830744

U2 - 10.1007/s11075-023-01740-5

DO - 10.1007/s11075-023-01740-5

M3 - Article

AN - SCOPUS:85182830744

SN - 1017-1398

JO - Numerical Algorithms

JF - Numerical Algorithms

ER -

An alternating direction implicit finite element Galerkin method for the linear Schrödinger equation

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