A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation

Abstract

The nonlinear Schrödinger equation is an important model equation in the study of quantum states of physical systems. To improve the computing efficiency, a fast algorithm based on the time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation is studied. The fourth-order compact difference scheme is used to approximate the spatial derivatives and the time two-mesh method is designed for efficiently solving the resulting nonlinear system. Comparing to the existing time two-mesh algorithm, the novelty of the new algorithm is that the fine mesh solution, which becomes available, is also used as the initial guess of the linear system, which can improve the calculation accuracy of fine mesh solutions. Compared to the two-grid finite element methods (or finite difference methods) for nonlinear Schrödinger equations, the numerical calculation of this method is relatively simple, and its two-mesh algorithm is implemented in the temporal direction. Taking advantage of the discrete energy, the result with O(τC4+τF2+h4) in the discrete L2-norm is obtained. Here, τC and τF are the temporal parameters on the coarse and fine mesh, respectively, and h is the space step size. Finally, some numerical experiments are conducted to demonstrate its efficiency and accuracy. The numerical results show that the new algorithm gives highly accurate results and preserves conservation laws of charge and energy. Furthermore, by comparing with the standard nonlinear implicit compact difference scheme, it can reduce the CPU time without loss of accuracy.

The cell-average based neural network for numerical approximation of the nonlinear Schrödinger equation

“…[19][20][21][22] employ the Galerkin method, which directly addresses the original control equation in integral form. Moreover, some comprehensive methods and other numerical methods can be found in [13,23,24] and [25][26][27][28][29][30][31][32], respectively. While classical numerical methods have yielded abundant results, there remains a need to develop a neural network approach to studying the NLS equation that excels in terms of stability, accuracy, and efficiency when compared to established classical numerical methods.…”

A Time Two-Mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

“…The TT-M technique was also employed by Niu et al [32] to develop a fast high-order compact difference scheme for the nonlinear distributed order fractional Sobolev model in porous media. Furthermore, He et al [33] extended the application of the TT-M method by studying a primary scheme of second-order convergence in time and fourth-order in space for solving the nonlinear Schrödinger equation with a time two-mesh high-order compact difference scheme. Despite the extensive research on the TT-M method in various fields, to the best of our knowledge, no study on the application of the TT-M method combined with finite difference to the SRLW equation has been discovered.…”

“…(i) Based on the time two-mesh technique, we proposed a scheme that achieves decoupling and the nonlinear term of the system is linearized by using Taylor's formula for a function with three variables, which is different from the literature [32,33]. In [32,33], the time two-mesh scheme is formulated by using Taylor's formula for a function with one or two variables. As a result, our scheme becomes a linearized system in approximate numerical solution and can reduce the computational time.…”

New Two-Level Time-Mesh Difference Scheme for the Symmetric Regularized Long Wave Equation

“…The combination of the time two-mesh (TT-M) technique [21][22][23][24][25][26][27] with other numerical methods also can improve the efficiency of solving nonlinear partial differential equations. Liu et al [21] investigated a finite element method with the TT-M technique, which was successfully applied to solve the fractional water wave model and other fractional models.…”

“…Liu et al [21] investigated a finite element method with the TT-M technique, which was successfully applied to solve the fractional water wave model and other fractional models. Afterward, other authors [22][23][24][25][26] used the TT-M method to study the numerical solutions for the partial differential equations such as the Allen-Cahn model, Sobolev model and the nonlinear Schrödinger equation. Gao et al [27] introduced a TT-M finite difference scheme for the SRLW equation, achieving first-order accuracy in time and second-order accuracy in space.…”