Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 395, Issue -, Pages 166-185Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2019.05.048
Keywords
Structure-preserving algorithm; Neumann boundary condition; Summation by parts operator; Scalar auxiliary variable approach
Funding
- National Natural Science Foundation of China [11771213, 61872422]
- National Key Research and Development Project of China [2016YFC0600310, 2018YFC0603500, 2018YFC1504205]
- Major Projects of Natural Sciences of University in Jiangsu Province of China [15KJA110002, 18KJA110003]
- Natural Science Foundation of Jiangsu Province, China [BK20171480]
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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 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.
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