A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations

On HSS-like iteration method for the space fractional coupled nonlinear Schrödinger equations

“…. , N − 1, and proved that the scheme (2.1) conserves the discrete mass and energy, and is unconditionally stable and convergent with order O(τ 2 + h 2 ) in the discrete l 2 norm [16,17]. By the initial boundary value conditions, we have…”

“…It was shown that a simple discretization scheme of the space fractional CNLS equations, even though implicit, leads to be unconditionally unstable. Recently, based on the fractional centered difference formula, in [15][16][17] the authors proposed an implicit conservative difference scheme which is unconditionally stable to discretize the space fractional CNLS equations. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-like matrix which is the diagonal-plus-Toeplitz matrix.…”

A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schrödinger equations

“…. , N -1, and it has been proved that scheme (2.1) conserves the discrete mass and energy, and is unconditionally stable and convergent with the order O(τ 2 + h 2 ) in the discrete l 2 -norm [22,23]. By the boundary values and initial conditions, we have…”

“…Moreover, the nonlocal property of the fractional operator resulted in dense or full matrices. Consequently, with the fractional centered difference formula, Wang et al [22,23] proposed recently an implicit conservative difference scheme, which is unconditionally stable, to discretize the spatial fractional CNLS equations. The coefficient matrix A of the discretized linear system about Eqs.…”

Galerkin finite element method for nonlinear fractional Schrödinger equations

“…For the time-fractional Schrödinger equations, Mohebbi et al [23] employed a meshless technique based on collocation methods and radial basis functions, Khan et al [16] derived approximating solutions by homotopy analysis methods, and Wei et al [35] gave discrete solution via a rigorous analysis of implicit fully discrete local discontinuous Galerkin method. For the space-fractional Schrödinger equations, some fully or linearly implicit difference methods were introduced and discrete conservation properties were analyzed in [30,31,33]. Two-dimensional problems were considered and a fourth-order ADI scheme was presented in [40].…”