Efficient numerical computation of time-fractional nonlinear Schrödinger equations in unbounded domain

Jiwei Zhang (1) , Dongfang Li (2) , Xavier Antoine (3, 4)

1 CSRC - Beijing Computational Science Research Center [Beijing]
2 HUST - Huazhong University of Science and Technology [Wuhan]
3 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
4 IECL - Institut Élie Cartan de Lorraine

The aim of this paper is to derive an efficient scheme for solving one-dimensional time-fractional nonlinear Schrödinger equations set in unbounded domains. We first derive some absorbing boundary conditions for the fractional system by using the unified approach introduced in [57,58] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized and the error estimate O(h 2 + τ) is stated. To accelerate the scheme in time, the fractional derivative is approximated through a linearized L1-scheme. Finally, we end the paper by some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme. In addition, we illustrate the behavior of the solution by reporting a few simulations for various parameter values of the fractional order 0 < α < 1, nonlinearities and potentials.

Mots clés: fractional PDE

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Jiwei Zhang, Dongfang Li, Xavier Antoine. Efficient numerical computation of time-fractional nonlinear Schrödinger equations in unbounded domain. Communications in Computational Physics, 2019, 25 (1), pp.218-243. ⟨10.4208/cicp.OA-2017-0195⟩. ⟨hal-01422725⟩