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Paper details
Number 4 - December 2019
Volume 29 - 2019
A conservative scheme with optimal error estimates for a multidimensional space-fractional Gross–Pitaevskii equation
Ahmed S. Hendy, Jorge E. Macías-Díaz
Abstract
The present work departs from an extended form of the classical multi-dimensional Gross–Pitaevskii equation, which
considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation.
It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check
that the generalized fractional model considered in this work also possesses conserved quantities, whence the development
of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross–Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and
show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover,
we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to
show that the method is capable of preserving the total mass and the total energy of the generalized system.
Keywords
generalized Gross–Pitaevskii system, Riesz fractional diffusion, discrete uniform Sobolev inequality, conservative method, optimal error bounds