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

TL;DR

This paper develops structure-preserving numerical algorithms for the 2D sine-Gordon equation with Neumann boundary conditions, using cell-centered and SBP-based discretizations, and combines them with symplectic and energy-preserving time integrators.

Contribution

It introduces two novel strategies for structure-preserving discretizations under Neumann boundary conditions, extending existing methods beyond zero or periodic boundaries.

Findings

Schemes preserve Hamiltonian structures and energy.

Higher-order accuracy achieved with SBP-based schemes.

Numerical experiments confirm effectiveness under Neumann conditions.

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 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.