Abstract
A compact finite difference (CFD) scheme is presented for the nonlinear Schrödinger equation involving a quintic term. The two discrete conservative laws are obtained. The unconditional stability and convergence in maximum norm with order O({tau }^{2}+h^{4}) are proved by using the energy method. A numerical experiment is presented to support our theoretical results.
Highlights
The Schrödinger (NLS) equation is one of the most important equations of mathematical physics with applications in many fields [1,2,3,4] such as plasma physics, nonlinear optics, water waves, and bimolecular dynamics
[28] Wang proposed a new difference scheme for NLS equation involving a quintic term and showed that convergence rates of the present scheme were of order O(τ 2 + h4)
For the exact solution of the initial-boundary value problem (1.1)–(1.3), we assume that max Un, δxUn, Un ∞ ≤ C
Summary
The Schrödinger (NLS) equation is one of the most important equations of mathematical physics with applications in many fields [1,2,3,4] such as plasma physics, nonlinear optics, water waves, and bimolecular dynamics. There are many studies on numerical approaches, including finite difference [5,6,7,8,9,10,11], finite element [12,13,14], and polynomial approximation methods [15, 16], of the initial or initial-boundary value problems of the Schrödinger equations. We consider the initial-boundary value problem for the NLS equation involving a quintic term:. Where u(x, t) is a complex function, f (x, t) is a real function, u0(x) is a prescribed smooth function, and i2 = –1
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