2019 **Abstract:** This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms are only valid for zero or periodic boundary conditions. The first strategy is based on the conventional second-order central difference quotient but with a cell-centered grid, while the other is established on the regular grid but incorporated with summation by parts (SBP) operators. Both the meth…

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(24 citation statements)

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“…Various numerical schemes such as finite difference schemes [9], two-level dissipationless Maxwell-Bloch systems [10], generalized leapfrog method [6], finite element methods [7], a split cosine scheme [11], a three-time level fourth-order explicit finite-difference scheme [12], method of lines [13], a modified predictor-corrector scheme [14], dual reciprocity boundary element method [15], a numerical method based on radial basis functions (RBFs) [16], boundary element method [4], meshless local Petrov-Galerkin method [17], a local weak meshless technique based on the radial point interpolation method [18], a method based on collocation and RBFs [19], meshless local boundary integral equation method [20], interpolated coefficient finite element method [21], differential quadrature methods [22][23][24][25], and space-time spectral collocation method [26] have been developed for solving the SG equation. Recently, localized methods of approximate particular solutions [27] and structure-preserving algorithms [28] have been sublimed for simulation of the SG equation.…”

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confidence: 99%

“…Various numerical schemes such as finite difference schemes [9], two-level dissipationless Maxwell-Bloch systems [10], generalized leapfrog method [6], finite element methods [7], a split cosine scheme [11], a three-time level fourth-order explicit finite-difference scheme [12], method of lines [13], a modified predictor-corrector scheme [14], dual reciprocity boundary element method [15], a numerical method based on radial basis functions (RBFs) [16], boundary element method [4], meshless local Petrov-Galerkin method [17], a local weak meshless technique based on the radial point interpolation method [18], a method based on collocation and RBFs [19], meshless local boundary integral equation method [20], interpolated coefficient finite element method [21], differential quadrature methods [22][23][24][25], and space-time spectral collocation method [26] have been developed for solving the SG equation. Recently, localized methods of approximate particular solutions [27] and structure-preserving algorithms [28] have been sublimed for simulation of the SG equation.…”

mentioning

confidence: 99%

“…The precision and capacity of the proposed algorithms are demonstrated by several 1D and 2D test problems. The main contribution of this article lies in the facts: (i) to the best of author's knowledge, barycentric rational interpolation is used first time for SG, (ii) semi-discrete problem is analyzed for truncation error and convergence analysis in L 2 and H 1 spaces, (iii) the two novel algorithms are compared to each other and already available techniques and found that the algorithms have accuracy and efficiency, and (iv) comparison point of view barycentric rational interpolation based algorithm (BRIA) and local radial basis functions based differential quadrature (LRBF-DQ) algorithms gives better accuracy than [15,16,19] and similar result to [4,15,16,18,20,22,27,28].…”

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confidence: 99%

“…More recently, inspired by the scalar auxiliary variable (SAV) approach [18,19], Cai et al developed a linearly implicit energy-conserving scheme for the sine-Gordon equation [1]. The resulting scheme leads to a linear system with constant coefficients that is easy to implement.…”

confidence: 99%

“…The IEQ method is an efficient way to construct linearly implicit energy-preserving schemes for the Hamiltonian partial differential equations (PDEs) [2,10,21]. About the idea of IEQ schemes, we refer to the review paper [23] and the references therein.…”

confidence: 99%