Dissipation-preserving spectral element method for damped seismic wave equations

Cai, Wenjun; Zhang, Huai; Wang, Yushun

doi:10.1016/j.jcp.2017.08.048

Cited by 6 publications

(2 citation statements)

References 36 publications

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“…In the study of [23], authors straightly applied this fact on the one-dimensional sine-Gordon equation and derived an energy-preserving integrator. A similar idea can be found in [24] for the simulation of two-dimensional seismic wave equations. While for the finite difference based schemes in [5], authors used Taylor expansions to achieve a consistent accuracy in all spatial grid points for semi-discretizations and further combined generalized AVF methods to obtain energy-preserving schemes with second and fourth orders, respectively.…”

Section: Introductionmentioning

confidence: 82%

Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

Cai

Jiang

Wang

et al. 2019

Journal of Computational Physics

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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 methodologies can provide conservative semi-discretizations with different forms of Hamiltonian structures and the discrete energy. However, utilizing the existing SBP formulas, schemes obtained by the second strategy can directly achieve higher-order accuracy while it is not obvious for schemes based on the cell-centered grid to make accuracy improved easily. Further combining the symplectic Runge-Kutta method and the scalar auxiliary variable (SAV) approach, we construct symplectic integrators and linearly implicit energy-preserving schemes for the two dimensional sine-Gordon equation, respectively. Extensive numerical experiments demonstrate their effectiveness with the homogeneous Neumann boundary conditions.

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Section: Introductionmentioning

confidence: 82%

Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

Cai

Jiang

Wang

et al. 2019

Journal of Computational Physics

Self Cite

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show abstract

“…So it is very meaningful to design a numerical method which can satisfy the structure of the original system, because it has conspicuous ability to preserve the geometric properties of phase space for a long time [12,15,22,24,26,32] and has good numerical stability [1]. For classical damped nonlinear Schrödinger equation, some conformal symplectic or conformal multisymplectic methods have been constructed [2,3,13,23]. However, the research on conformal symplectics and multisymplectics of fractional differential equations is relatively few and we notice that the analysis of convergence is almost absent for conformal multisymplectic methods.…”

Section: Introductionmentioning

confidence: 99%

Conformal structure‐preserving method for two‐dimensional damped nonlinear fractional Schrödinger equation

Ding

2023

Numerical Methods Partial

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This paper mainly analyzes conservation laws and convergence of splitting conformal multisymplectic scheme for solving the two-dimensional damped nonlinear fractional Schrödinger equation. In order to obtain conformal multisymplectic scheme, first using Strang splitting method, the original problem is split into conservative multisymplectic and dissipative multisymplectic systems. The conservative multisymplectic system is numerically solved and the dissipative part is solved exactly. And then there shows the corresponding conservation laws and numerical scheme, in which the implicit midpoint method is used in time and the Fourier pseudospectral method is used in space, it also preserves conformal multisymplectic, the global conformal symplectic and global conformal mass conservation laws. Most important of all, we discuss convergence of the proposed scheme which is second-order accuracy in time and spectral accuracy in space. Finally, the validity and accuracy of the theoretical results are verified by several numerical examples.

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Dissipation-Preserving Rational Spectral-Galerkin Method for Strongly Damped Nonlinear Wave System Involving Mixed Fractional Laplacians in Unbounded Domains

Guo

Yan²,

Li³

et al. 2022

J Sci Comput

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Dissipation-preserving spectral element method for damped seismic wave equations

Cited by 6 publications

References 36 publications

Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

Conformal structure‐preserving method for two‐dimensional damped nonlinear fractional Schrödinger equation

Dissipation-Preserving Rational Spectral-Galerkin Method for Strongly Damped Nonlinear Wave System Involving Mixed Fractional Laplacians in Unbounded Domains

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