An hr-adaptive method for the cubic nonlinear Schrödinger equation

J.A. Mackenzie, W.R. Mekwi

Research output: Contribution to journalArticle

Abstract

The nonlinear Schr¨odinger 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. Mesh refinement methods (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.
Original languageEnglish
Article number112320
JournalJournal of Computational and Applied Mathematics
Volume364
Early online date9 Jul 2019
DOIs
Publication statusE-pub ahead of print - 9 Jul 2019

Fingerprint

Nonlinear equations
Schrodinger equation
Optical fiber communication
Quantum theory
Biomolecules
Physics
Plasmas

Keywords

  • Adaptivity
  • Moving mesh method
  • hr-adaptivity
  • Cubic nonlinear Schrodinger 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

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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 Schr¨odinger 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. Mesh refinement methods (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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