Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions
It extends structure-preserving methods to a new boundary condition type for the sine-Gordon equation, but the approach is incremental as it adapts existing techniques.
This paper develops structure-preserving algorithms for the two-dimensional sine-Gordon equation with homogeneous Neumann boundary conditions, which were previously limited to zero or periodic boundaries. Two strategies are proposed: one using cell-centered grids and another using summation-by-parts operators, both achieving conservative semi-discretizations, with the latter enabling higher-order accuracy.
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.