A compact split step Padé scheme for higher-order nonlinear Schrödinger equation (HNLS) with power law nonlinearity and fourth order dispersion

Finite difference scheme for a high order nonlinear Schrodinger equation with localized damping

“…We have also compared our proposal with the one from Smadi and Bahloul, observing an evident difference between both outputs. We deduce that our numerical method adapts better and more efficiently to the numerical resolution of the HNSL equation with respect to the known methods in the literature (see Smadi and Bahloul [19]), for various examples, with or without damping. Our method can also compute reasonable results using a small computer time at a home PC.…”

“…The only known work dealing with numerical methods for Problem (1.1) was proposed by Smadi and Bahloul [19] (from now on: SB scheme), which was implemented again in [20] with other examples. Their scheme splits Problem (1.1) in two parts: a linear one, which is solved using a Compact Finite Difference scheme [13]; and a nonlinear part, which solves the problem through an explicit fourth-order Runge-Kutta method.…”

“…Remark 3.1. The paper of Smadi and Bahloul [19] does not give details regarding the discretization of the first-order derivatives in non-linear terms (the description of the Runge-Kutta method is presented in [19] in terms of a f (u), and not of a f (u, u x )). We assume in our simulations a finite deference centered as an approximation of u x , however, it is worth noting that we tested with other more efficient approaches obtaining very similar results for the SB method that we show here.…”

“…The higher-power term u 4 u x was rewritten as a linear combination of other derivatives in order to obtain specific conservation properties. Smadi and Bahloul [19] [20] combined a Compact Padé Finite Difference scheme [13] with a fourth order Runge-Kutta (RK4) scheme. They splitted the problem in two parts: a linear one which is solved using the finite difference scheme; and a nonlinear, which is solved using the RK4 scheme.…”

A Padé compact high-order finite volume scheme for nonlinear Schrödinger equations

“…Thus, compact schemes are a good option for simulation of wave dominated problems [12,28]. Some researchers employed compact finite difference methods for the numerical solution of the nonlinear Schrödinger equations [5,19,25]. In contrast, few of studies [9,18,11] were conducted on the solution of the CNLS equation by compact finite volume schemes.…”