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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:395

Article

Cai, Wenjun¹  Jiang, Chaolong²  Wang, Yushun¹  Song, Yongzhong¹ 

[1] Nanjing Normal Univ, Sch Math Sci, Jiangsu Key Lab NSLSCS, Nanjing, Jiangsu, Peoples R China

[2] Yunnan Univ Finance & Econ, Sch Stat & Math, Kunming, Yunnan, Peoples R China

关键词: Structure-preserving algorithm;    Neumann boundary condition;    Summation by parts operator;    Scalar auxiliary variable approach;   

DOI  :  10.1016/j.jcp.2019.05.048

来源: Elsevier

【 摘 要 】

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 implicit midpoint 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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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