Several conservative compact schemes for a class of nonlinear Schrödinger equations with wave operator

Abstract

In this paper, several different conserving compact finite difference schemes are developed for solving a class of nonlinear Schrödinger equations with wave operator. It is proved that the numerical solutions are bounded and the numerical methods can achieve a convergence rate of O(τ 2 + h 4 ) in the maximum norm. Moreover, by applying Richardson extrapolation, the proposed methods have a convergence rate of O(τ 4 + h 4 ) in the maximum norm. Finally, several numerical experiments are presented to illustrate the theoretical results. MSC: 65N06; 65N12

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

“…where 𝜚 𝜂,𝜎 i is as given in (6). Using the orthogonality relation of T 𝜂,𝜎 i (x)} N i=0 , we obtain that…”

“…There are various results concerning the numerical simulation of the nonlinear Schrödinger equation and its variants. Such results include finite difference method, 1–6 finite element method, 7–10 spectral collocation methods, 11–15 and meshless technique 16–19 …”

“…There are various results concerning the numerical simulation of the nonlinear Schrödinger equation and its variants. Such results include finite difference method, [1][2][3][4][5][6] finite element method, [7][8][9][10] spectral collocation methods, [11][12][13][14][15] and meshless technique. [16][17][18][19] To the best of our knowledge, there have only been a few results on the numerical solution of the Schrödinger equation in unbounded domain.…”

A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator

“…In the past several years, various conservative and accurate numerical methods have been developed for the NLSW, including spectral methods [15,23], finite element methods [5,9], finite difference methods [2,6,16,17,25,29,30] and so on. For the FNLSW, to the authors' knowledge, the literature limited.…”

Nonlinear conservation laws for the Schrödinger boundary value problems of second order

“…Regarding the existence of a solution of the problem related to the Schrödinger operator (1.1) we refer to [33] and [24], respectively, for the evolutionary dam problem with homogeneous coefficients and for a class of free boundary problem with respect to the Schrödinger operator in heterogeneous domain. The regularity of the solution of the problem [24] with respect to the Schrödinger operator was discussed in [34] (see also [13,20,21,31,36]), where it was proved that ω ∈ C 0 ([0, L]; L p ‫))ג(‬ for all p ∈ [1, +∞) in the class of free boundary problems with respect to the Schrödinger operator of types 7) and that f ∈ C 0 ([0, M]; L p ‫))ג(‬ for all p ∈ [1,2] in the second-order class. More results as regards Schrödinger-type equations, wavelet analysis, distribution theory and calculus of variations were studied in previous work [16,18,19,27,32,50].…”