Theoretical analysis of a conservative finite-difference scheme to solve a Riesz space-fractional Gross–Pitaevskii system

A Mass- and Energy-Conserving Numerical Model for a Fractional Gross–Pitaevskii System in Multiple Dimensions

“…A Matlab implementation of the numerical model is provided in the Appendix of this work, and we used to produce approximations to the solutions of our continuous model. The results show that the energy and mass are approximately constant, in agreement with the theoretical results derived in this work, and improving computationally efforts already reported [20,26].…”

“…It is worth pointing out that there are already some reports by these same authors on the design of numerical methods to solve double Gross-Pitaevskii-type systems with fractional diffusion, most notably the papers by Serna-Reyes et al [26] and Serna-Reyes and Macías-Díaz [20]. In both works, the same system of fractional partial differential equations is investigated using a finite-difference methodology, the fractional derivatives are of the Riesz type, and they are discretized using fractional-order centered differences.…”

“…In both works, the same system of fractional partial differential equations is investigated using a finite-difference methodology, the fractional derivatives are of the Riesz type, and they are discretized using fractional-order centered differences. To start with, the proofs for the existence of solutions of the numerical model are more complicated in the case of the published papers [20,26]. Indeed, the arguments in those cases require using some fixed-point theorem in view of the nonlinear nature of those works.…”

“…Finally, in terms of implementation, the present numerical methodology is easy to implement computationally, in view that the code only requires solving a couple of linear systems at each time steps. On the other hand, the finite-difference schemes reported in [20,26] are harder to code. This is again due to the nonlinear nature of the numerical models.…”

An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System

“…In particular, there have been substantial developments in the theory and application of numerical methods for fractional partial differential equations. For example, from a theoretical point of view, theoretical analyses of conservative finitedifference schemes to solve the Riesz space-fractional Gross-Pitaevskii system have been proposed in the literature [4], along with convergent three-step numerical methods to solve double-fractional condensates, explicit dissipation-preserving methods for Riesz space-fractional nonlinear wave equations in multiple dimensions [5], energy conservative difference schemes for nonlinear fractional Schrödinger equations [6], conservative difference schemes for the Riesz space-fractional sine-Gordon equation [7], high-order central difference schemes for Caputo fractional derivatives [8], among other examples.…”