An hr-adaptive method for the cubic nonlinear Schrödinger equation

J.A. MacKenzie, W.R. Mekwi

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

The nonlinear Schrödinger equation (NLSE) is one of the most important equations in quantum mechanics, and appears in a wide range of applications including optical fibre communications, plasma physics and biomolecule dynamics. It is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients. Adaptive moving mesh methods (r-adaptive) attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features. This approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest. Adaptive mesh refinement (h-adaptive), where the mesh is locally coarsened or refined, is an alternative adaptive strategy which is popular for time-independent problems. In this paper, we consider the effectiveness of a combined method (hr-adaptive) to solve the NLSE in one space dimension. Simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches. The method is also shown to control the spatial error based on the user's input error tolerance. Evidence is also presented indicating second-order spatial convergence using a novel monitor function to generate the adaptive moving mesh.

LanguageEnglish
Article number112320
Number of pages20
JournalJournal of Computational and Applied Mathematics
Volume364
Early online date9 Jul 2019
DOIs
Publication statusE-pub ahead of print - 9 Jul 2019

Fingerprint

Adaptive Method
Nonlinear equations
Nonlinear Schrödinger Equation
Moving Mesh
Adaptive Mesh
Moving Mesh Method
Optical Fiber Communication
Plasma Physics
Adaptive Mesh Refinement
Adaptive Strategies
Biomolecules
Combined Method
Optical fiber communication
Quantum Mechanics
Tolerance
Quantum theory
Resolve
Monitor
Optimise
Mesh

Keywords

  • adaptivity
  • moving mesh methods
  • hr-adaptivity
  • cubic nonlinear Schrödinger equation

Cite this

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An hr-adaptive method for the cubic nonlinear Schrödinger equation. / MacKenzie, J.A.; Mekwi, W.R.

In: Journal of Computational and Applied Mathematics, Vol. 364, 112320, 15.01.2020.

Research output: Contribution to journalArticle

TY - JOUR

T1 - An hr-adaptive method for the cubic nonlinear Schrödinger equation

AU - MacKenzie, J.A.

AU - Mekwi, W.R.

PY - 2019/7/9

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AB - The nonlinear Schrödinger equation (NLSE) is one of the most important equations in quantum mechanics, and appears in a wide range of applications including optical fibre communications, plasma physics and biomolecule dynamics. It is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients. Adaptive moving mesh methods (r-adaptive) attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features. This approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest. Adaptive mesh refinement (h-adaptive), where the mesh is locally coarsened or refined, is an alternative adaptive strategy which is popular for time-independent problems. In this paper, we consider the effectiveness of a combined method (hr-adaptive) to solve the NLSE in one space dimension. Simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches. The method is also shown to control the spatial error based on the user's input error tolerance. Evidence is also presented indicating second-order spatial convergence using a novel monitor function to generate the adaptive moving mesh.

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KW - hr-adaptivity

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UR - https://arxiv.org/abs/1907.02472

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