Semi-explicit symplectic partitioned Runge–Kutta Fourier pseudo-spectral scheme for Klein–Gordon–Schrödinger equations

Kong, Linghua; Zhang, Jingjing; Cao, Ying; Duan, Yuhua; Huang, Hong

doi:10.1016/j.cpc.2010.04.003

Cited by 55 publications

(22 citation statements)

References 17 publications

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“…The nonlinear Klein-Gordon equation plays a significant role in many scientific fields such as nonlinear optics, solid state physics, fluid dynamics and quantum field theory. To examine these types of equations, scientists have used many effective techniques such as the auxiliary equation method [23], pseudo-spectral method [26], Jacobi elliptic functions [17] and the tanh-sech method [28].…”

Section: Necdet Bi̇ldi̇k and Sinan Deni̇zmentioning

confidence: 99%

New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques

Bıldık¹,

Deni̇z²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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In this study, we present the new approximate solutions of the nonlinear Klein-Gordon equations via perturbation iteration technique and newly developed optimal perturbation iteration method. Some specific examples are given and obtained solutions are compared with other methods and analytical results to confirm the good accuracy of the proposed methods.We also discuss the convergence of the optimal perturbation iteration method for partial differential equations. The results reveal that perturbation iteration techniques,unlike many other techniques in literature, converge rapidly to exact solutions of the given problems at lower order of approximations.

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Section: Necdet Bi̇ldi̇k and Sinan Deni̇zmentioning

confidence: 99%

New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques

Bıldık¹,

Deni̇z²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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“…Later on, various structure-preserving Fourier pseudospectral methods were developed (e.g., see Refs. [28][29][30][31]). However, there are few conformal Fourier pseudospectral schemes and the error estimate of the conformal Fourier pseudospectral schemes is not valid to our knowledge.…”

Section: Introductionmentioning

confidence: 99%

Optimal error estimate of a conformal Fourier pseudo‐spectral method for the damped nonlinear Schrödinger equation

Jiang

Cai

Wang

2018

Numerical Methods Partial

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In this article, a Fourier pseudospectral method, which preserves the conforal conservation la, is proposed for solving the damped nonlinear Schrödinger equation. Based on the energy method and the semi‐norm equivalence between the Fourier pseudospectral method and the finite difference method, a priori estimate for the new method is established, which shows that the proposed method is unconditionally convergent with order of O ( normalτ 2 + J 1 − r ) in the discrete L ∞ ‐norm, where normalτ is the time step and J is the number of collocation points used in the spectral method. Some numerical results are addressed to confirm our theoretical analysis.

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“…We found that the scheme can be adopted to solve (1.1)–(1.4) by careful analysis of the effect between the interaction of the coupled terms. Here, we remark that for the classical KGS equation (with derivatives of integer order), linearized (semilinearized) schemes can be established directly because the second‐order derivative

ϕ_{t t}

can be approximated by the standard central difference operator

(ϕ^{n + 1} - 2 ϕ^{n} + ϕ^{n - 1}) / {normalτ}^{2}

at the time level

n

(thus the nonlinear term

| {normalψ}^{n} |^{2}

can be evaluated at the level

n

) , where

normalτ

is the time step size. However, the situation is different and more complicate for the fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

“…The coupled KGS equation describes interaction between conservative complex neutron field and neutral meson Yukawa in quantum field theory. The equation and its variants have been studied in both theoretical and numerical aspects by many researchers in recent years, see [1][2][3][4][5][6][7] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

Lyu

Vong

2018

Numerical Methods Partial

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In this work, we study finite difference scheme for coupled time fractional Klein‐Gordon‐Schrödinger (KGS) equation. We proposed a linearized finite difference scheme to solve the coupled system, in which the fractional derivatives are approximated by some recently established discretization formulas. These formulas approximate the solution with second‐order accuracy at points different form the grid points in time direction. Taking advantage of this property, our proposed linearized scheme evaluates the nonlinear terms on the previous time level. As a result, iterative method is dispensable. The coupled terms in the scheme bring difficulties in analysis. By carefully studying these effects, we proved that the proposed scheme is unconditionally convergent and stable in discrete L 2 norm with energy method. Numerical results are included to justify the theoretical statements.

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Semi-explicit symplectic partitioned Runge–Kutta Fourier pseudo-spectral scheme for Klein–Gordon–Schrödinger equations

Cited by 55 publications

References 17 publications

New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques

New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques

Optimal error estimate of a conformal Fourier pseudo‐spectral method for the damped nonlinear Schrödinger equation

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

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