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Journal of Computational Mathematics

Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library
Christophe Besse1; Romain Duboscq2; Stefan Le Coz1
1 Institut de Mathématiques de Toulouse ; UMR5219, Université de Toulouse ; CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France
2 Institut de Mathématiques de Toulouse ; UMR5219, Université de Toulouse ; CNRS, INSA IMT, F-31077 Toulouse, France
The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 1-47.

Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schrödinger equation. Whereas in the last ten years they have known considerable developments on the theoretical side, their study from the numerical point of view remains in its early stages. The goal of this paper is to present the Grafidi library, a Python library which has been developed with the numerical simulation of nonlinear Schrödinger equations on graphs in mind. We will show how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. We will also simulate the dynamics of the nonlinear Schrödinger equation with a Crank-Nicolson relaxation scheme and a Strang splitting scheme. Finally, in a series of numerical experiments on various types of graphs, we will compare the outcome of our numerical calculations for ground states with the existing theoretical results, thereby illustrating the versatility and efficiency of our implementations in the framework of the Grafidi library.

Published online:
DOI: 10.5802/smai-jcm.78
Classification: 35R02, 65N06, 35Q55
Keywords: Quantum Graphs, Python Library, Nonlinear Schrodinger equation, Finite Differences, Ground states
Christophe Besse 1; Romain Duboscq 2; Stefan Le Coz 1

1 Institut de Mathématiques de Toulouse ; UMR5219, Université de Toulouse ; CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France
2 Institut de Mathématiques de Toulouse ; UMR5219, Université de Toulouse ; CNRS, INSA IMT, F-31077 Toulouse, France
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Christophe Besse; Romain Duboscq; Stefan Le Coz. Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library. The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 1-47. doi : 10.5802/smai-jcm.78. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.78/

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