Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations

Abstract

In the manuscript, we present several numerical schemes to approximate the coupled nonlinear Schrödinger equations. Three of them are high-order compact and conservative, and the other two are noncompact but conservative. After some numerical analysis, we can find that the schemes are uniquely solvable and convergent. All of them are conservative and stable. By calculating the complexity, we can find that the compact schemes have the same computational cost with the noncompact ones. Numerical illustrations support our analysis. They verify that compact schemes are more efficient than noncompact ones from computation cost and accuracy.

Computational Methods for Three Coupled Nonlinear Schrödinger Equations

“…In order to develop a numerical method for solving the system given in (12) (20) is of second order accuracy in both directions space and time, and it is unconditionally stable using von-Neumann stability analysis, see Ismail [8]. A nonlinear block tridiagonal system must be solved at each time step.…”

“…Finite difference and finite element methods are used to solve this system by Ismail [5]- [10]. A conservative compact finite difference schemes are given in [11] [12]. In [3] [4], Xing Lü studied the bright soliton collisions with shape change by intensity for the coupled Sasa-Satsuma system in the optical fiber communications.…”

Efficient energy‐preserving scheme of the three‐coupled nonlinear Schrödinger equation

“…Some authors also paid their attentions to two‐component CNLS equation. () The first author proposed some efficient energy‐preserving schemes for it . Ma et al developed a new kind of multisymplectic integrators which focused on the advantages of high‐order compact method, splitting technique, and multisymplectic integrator .…”

“…It features high accuracy, small stencil, structure‐preserving, low dissipating. () In the following, we focus on developing energy‐preserving schemes for the T‐CNLS equation by combing the AVF method and high‐order compact method. To the goal, we first recast the T‐CNLS equation into the Hamiltonian frame.…”

Optimal error estimate of two linear and momentum‐preserving Fourier pseudo‐spectral schemes for the RLW equation

“…Numerical methods that preserve at least some of the structural properties of the continuous dynamical system are called geometric integrators or structure-preserving algorithms [22][23][24]. In [28][29][30][31][32], some methods that conserve energy conservation laws were developed. Sun and Qin [25] constructed a multisymplectic Preissman scheme by using the implicit midpoint rule both in space and time.…”