Efficient Numerical Computation of Time-Fractional Nonlinear Schrödinger Equations in Unbounded Domain

Abstract

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrödinger equation set in an unbounded domain. We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in [47, 48] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized, a stability analysis is developed and the error estimate O(h 2 +τ) is stated. To accelerate the L1-scheme in time, a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative. The resulting algorithm is highly efficient for long time simulations. Finally, we end the paper by reporting some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme.

On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross–Pitaevskii equations

“…for 1 ≤ j ≤ J − 1. As discussed in [84], the convergence orders of the schemes (4.39) and (4.40) are respectively O(h 2 + ∆t) and O(h 2 + ∆t + ε), if the solution ψ is smooth enough. We point out that the choice of ε is far smaller than (h 2 +∆t) to not affect the convergence order.…”

“…For the Fourier pseudospectral (SP) scheme (see subsection 3.2), we will use a periodic BC. In addition, if the solution ψ is not confined in D, i.e., ψ can strike the boundary, then much more complicated BCs are required, such as the transparent, artificial and absorbing BCs as well as perfectly matched layers (see references [4,9,86] for the integer order Schrödinger equations and [84] for fractional order problems). However, this is out of the scope of the current paper.…”

Towards Perfectly Matched Layers for time-dependent space fractional PDEs

“…[22,23]), generalizing in some sense the notion of FPDEs. Let us remark that such boundary conditions were also nontrivially derived for time-fractional PDEs [32,47,49].…”

An Implicit Solver for the Time-Dependent Kohn-Sham Equation

“…There have been lots of numerical methods in the market to solve the TDKS equation in the time domain, people may refer to [3,7,14] and references therein for detail. People may also refer to [11,16,28] for numerical methods of Schrödinger equation. Among those grid-based numerical methods, the finite difference methods [1], the finite element methods [3,8,9,17,18,27], the discontinuous Galerkin methods [20], the wavelet methods [12] etc.…”