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.
In matrix theory and numerical analysis there are two very famous and important results. One is Geršgorin circle theorem, the other is strictly diagonally dominant theorem. They have important application and research value, and have been widely used and studied. In this paper, we investigate generalized diagonally dominant matrices and matrix eigenvalue inclusion regions. A class of G-function pairs is proposed, which extends the concept of G-functions. Thirteen kind of G-function pairs are established. Their properties and characteristics are studied. By using these special G-function pairs, we construct a large number of sufficient and necessary conditions for strictly diagonally dominant matrices and matrix eigenvalue inclusion regions. These conditions and regions are composed of different combinations of G-function pairs, deleted absolute row sums and column sums of matrices. The results extend, include and are better than some classical results.
In this paper, we investigate the preconditioned AOR method for solving linear systems. We study two general preconditioners and propose some lower triangular, upper triangular and combination preconditioners. For A being an L-matrix, a nonsingular M-matrix, an irreducible L-matrix and an irreducible nonsingular M-matrix, four types of comparison theorems are presented, respectively. They contain a general comparison result, a strict comparison result and two Stein-Rosenberg type comparison results. Our theorems include and are better than almost all known corresponding results.
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