A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems

Agissilaos Athanassoulis; Theodoros  Katsaounis; Irene Kyza; Stephen Metcalfe

doi:10.1016/j.jcp.2023.112307

A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems

Agissilaos Athanassoulis (Lead / Corresponding author), Theodoros Katsaounis, Irene Kyza (Lead / Corresponding author), Stephen Metcalfe

Mathematics

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

2 Citations (Scopus)

41 Downloads (Pure)

Abstract

We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of [10], [11], [34] for the nonlinear Schrödinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.

Original language	English
Article number	112307
Number of pages	18
Journal	Journal of Computational Physics
Volume	490
Early online date	22 Jun 2023
DOIs	https://doi.org/10.1016/j.jcp.2023.112307
Publication status	Published - 1 Oct 2023

Keywords

Schrödinger-Poisson system
energy preserving scheme
relaxation scheme in time
finite element method
Finite element method
Energy preserving scheme
Relaxation scheme in time

ASJC Scopus subject areas

Computational Mathematics
General Physics and Astronomy
Applied Mathematics
Numerical Analysis
Computer Science Applications
Modelling and Simulation
Physics and Astronomy (miscellaneous)

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10.1016/j.jcp.2023.112307Licence: CC BY

Final published versionFinal published version, 1.07 MBLicence: CC BY

Cite this

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title = "A novel, structure-preserving, second-order-in-time relaxation scheme for Schr{\"o}dinger-Poisson systems",

abstract = "We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schr{\"o}dinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of [10], [11], [34] for the nonlinear Schr{\"o}dinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.",

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author = "Agissilaos Athanassoulis and Theodoros Katsaounis and Irene Kyza and Stephen Metcalfe",

note = "Funding Information: The authors acknowledge the support from the Carnegie Trust Research Incentive Grant RIG008215. I.K. would also like to acknowledge the support from London Mathematical Society through an Emmy Noether Fellowship. In addition, Th. K. and I.K. thank the Edinburgh Mathematical Society for the Covid Recovery Fund that allowed for the completion and the submission of this paper. Copyright: {\textcopyright} 2023 The Author(s). Published by Elsevier Inc.",

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A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems. / Athanassoulis, Agissilaos (Lead / Corresponding author); Katsaounis, Theodoros ; Kyza, Irene (Lead / Corresponding author) et al.
In: Journal of Computational Physics, Vol. 490, 112307, 01.10.2023.

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

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T1 - A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems

AU - Athanassoulis, Agissilaos

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AU - Kyza, Irene

AU - Metcalfe, Stephen

N1 - Funding Information: The authors acknowledge the support from the Carnegie Trust Research Incentive Grant RIG008215. I.K. would also like to acknowledge the support from London Mathematical Society through an Emmy Noether Fellowship. In addition, Th. K. and I.K. thank the Edinburgh Mathematical Society for the Covid Recovery Fund that allowed for the completion and the submission of this paper. Copyright: © 2023 The Author(s). Published by Elsevier Inc.

PY - 2023/10/1

Y1 - 2023/10/1

N2 - We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of [10], [11], [34] for the nonlinear Schrödinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.

AB - We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of [10], [11], [34] for the nonlinear Schrödinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.

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KW - Relaxation scheme in time

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A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems

Abstract

Keywords

ASJC Scopus subject areas

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Covid Recovery Grant

LMS Emmy Noether Fellowship

Efficient Numerical Algorithms for the Schrodinger- Poisson System with Applications in Cosmology