Abstract
Combining the compact method with the structure-preserving algorithm, we propose a compact local energy-preserving scheme and a compact local momentum-preserving scheme for the nonlinear Schrödinger equation with wave operator (NSEW). The convergence rates of both schemes are Oh4+τ2. The discrete local conservative properties of the presented schemes are derived theoretically. Numerical experiments are carried out to demonstrate the convergence order and local conservation laws of the developed algorithms.
Highlights
Introduction e nonlinearSchrodinger equation with wave operator (NSEW) is a very important model in mathematical physics with applications in a wide range, such as plasma physics, water waves, nonlinear optics, and bimolecular dynamics [1, 2]
Several numerical algorithms have been studied for solving the nonlinear Schrodinger equation with wave operator (NSEW) (Refs. [3,4,5,6,7,8,9,10,11] and references therein)
Structure-preserving algorithms were proposed to solving the Hamiltonian systems [12,13,14,15] and applied to various PDEs, such as the nonlinear Schrodingertype equation [16,17,18,19], wave equation [20], and KdV equation [21]. e important feature of the structurepreserving algorithm is that it can maintain certain invariant quantities and has the ability of long-term simulation
Summary
It is not difficult to obtain the following:. (iii) Discrete Leibniz rule: Dx(f · g)j θfj+1 +(1 − θ)fj · Dxgj +(1 − θ)gj+1 + θgj · Dxfj, Dt(f · g)n θfn+1 +(1 − θ)fn · Dtgn +(1 − θ)gn+1 + θgn · Dtfn,. Θ 1, Dx(f · g)i fi+1 · Dxgi + Dxfi · gi. We get Dxδ−x1fgj δ−x1fj+1 · Dxgj + Dxδ−x1fj · gj. VN)T, v0 vN, we define the inner product and norms as. ⎧⎨ ptt − pxx − αqt + β p2 + q2p 0, ⎩ qtt − qxx + αpt + β p2 + q2q 0. Letting pt ξ, qt η, px ω, and qx ], system (14) can be written as.
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