Multi-Symplectic Fourier Pseudospectral Method for the Kawahara Equation

Gong, Yuezheng; Cai, Jiaxin; Wang, Yushun

doi:10.4208/cicp.090313.041113a

Cited by 50 publications

(27 citation statements)

References 22 publications

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“…Now, we present some Lemmas which play an important role in numerical analysis and numerical implementation for (2.10). But before that we need to introduce two new seminorms over

{bold-italicX}_{h}

| u |_{h} = {〈 - {bold-italicD}_{2} u, u 〉}_{J}^{\frac{1}{2}}, ‖ {bold-italicD}_{1} u ‖_{J} = {〈 {bold-italicD}_{1} u, {bold-italicD}_{1} u 〉}_{J}^{\frac{1}{2}} = {〈 - {bold-italicD}_{1}^{2} u, u 〉}_{J}^{\frac{1}{2}} .

Lemma () For the matrices

bold-italicB

{bold-italicD}_{1}

and

{bold-italicD}_{2}

, the following results hold

B = F^{- 1} Λ_{1} F,

{bold-italicD}_{1} = F^{- 1} Λ_{2} F,

{bold-italicD}_{2} = F^{- 1} Λ_{3} F,

where

Λ_{1}

Λ_{2}

, and

Λ_{3}

are the diagonal matrices whose (non‐zero) entries are the scaled wave‐numbers

Λ_{1} = <...>

…”

Section: Fourier Pseudospectral Methodsmentioning

confidence: 99%

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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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“…Now, we present some Lemmas which play an important role in numerical analysis and numerical implementation for (2.10). But before that we need to introduce two new seminorms over

{bold-italicX}_{h}

| u |_{h} = {〈 - {bold-italicD}_{2} u, u 〉}_{J}^{\frac{1}{2}}, ‖ {bold-italicD}_{1} u ‖_{J} = {〈 {bold-italicD}_{1} u, {bold-italicD}_{1} u 〉}_{J}^{\frac{1}{2}} = {〈 - {bold-italicD}_{1}^{2} u, u 〉}_{J}^{\frac{1}{2}} .

Lemma () For the matrices

bold-italicB

{bold-italicD}_{1}

and

{bold-italicD}_{2}

, the following results hold

B = F^{- 1} Λ_{1} F,

{bold-italicD}_{1} = F^{- 1} Λ_{2} F,

{bold-italicD}_{2} = F^{- 1} Λ_{3} F,

where

Λ_{1}

Λ_{2}

, and

Λ_{3}

are the diagonal matrices whose (non‐zero) entries are the scaled wave‐numbers

Λ_{1} = <...>

…”

Section: Fourier Pseudospectral Methodsmentioning

confidence: 99%

“…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 section, we will apply Fourier pseudospectral method to the multi-symplectic Hamiltonian system (2.17) with periodic boundary condition. Fourier pseudospectral method [32][33][34][35][36][37][38][39] J.J. WANG have been proven very powerful for periodic initial value problems with constant coefficients. In [32], Chen and Qin proposed multi-symplectic Fourier pseudospectral method for integrate nonlinear Schrödinger equation with periodic boundary condition.…”

Section: Introductionmentioning

confidence: 99%

“…In [32], Chen and Qin proposed multi-symplectic Fourier pseudospectral method for integrate nonlinear Schrödinger equation with periodic boundary condition. Wang [34] made some numerical analysis for the nonlinear Schrödinger equation.Gong [36] et al proposed multi-symplectic Fourier pseudospectral method for Kawahara equation with periodic boundary condition.…”

Section: Introductionmentioning

confidence: 99%

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Multi-symplectic Fourier Pseudospectral Method for a Hight Order Wave Equation of KdV Type

Wang¹

2015

JCM

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The higher order wave equation of KdV type, which describes many important physical phenomena, has been investigated widely in last several decades. In this work, multisymplectic formulations for the higher order wave equation of KdV type are presented, and the local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. The multi-symplectic discretization of each formulation is calculated by the multi-symplectic Fourier pseudospectral scheme. Numerical experiments are carried out, which verify the efficiency of the Fourier pseudospectral method.

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Efficient energy‐preserving methods for the Schrödinger‐Boussinesq equation

Yan¹,

Zheng

et al. 2022

Math Methods in App Sciences

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In this paper, we present two novel energy‐preserving schemes for the Schrödinger‐Boussinesq system based on the second‐order Adams prediction‐correction method. We first introduce supplementary variables to the initial model simultaneous its invariants to obtain a consistent and structural stable system. Then, we discretize the extended system by directly using the Fourier pseudo‐spectral method in space and second‐order Adams prediction‐correction method in time to construct conservative schemes. Numerical results show that the proposed schemes can reach second‐order accuracy and can conserve the discrete invariants very well.

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Multi-Symplectic Fourier Pseudospectral Method for the Kawahara Equation

Cited by 50 publications

References 22 publications

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

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

Multi-symplectic Fourier Pseudospectral Method for a Hight Order Wave Equation of KdV Type

Efficient energy‐preserving methods for the Schrödinger‐Boussinesq equation

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