New Algorithm For Solution Of Periodic Block Tridiagonal Linear Algebraic Equations

The general elliptic partial differential equation with periodic boundary conditions is as followsSuppose that there is a rectangular mesh, from Evans' discretization method, the following difference scheme is obtainedIt can be written in matrix notation as Au = s, where A is periodic block-tridiagonal matrix.Moreover, using a combined compact difference scheme proposed by Nihei and Ishii and discretizing the shallow water equationsWe can also get Au = s whose coefficient matrix is large periodic block-tridiagonal matrix.In this paper, several new iterative methods and direct methods for solving linear system whose coefficient matrix is large periodic block-tridiagonal matrix are proposed. They are compared with some existing methods. The main work is as follows(1) By incompletely LU triangular decomposition of the coefficientmatrix linear PE method and quadratic PE method are obtained. Then we propose linear PEk method and quadratic PE* method through introducing a parameter k. Linear PE method (i.e. PE method) and Linear PEk method (i.e. PEk method) are mainly discussed in the paper. Under the condition that the coefficient matrix is Hermite positive definite matrix, M-matrix or H-matrix, the solvabilities and convergences of the two methods are proved. Some examples are given to illustrate that their convergences are better than some existing methods (such as block Jacobi method and SBGS method).(2) Present two direct methods for solving linear system whose coefficient matrix is large periodic block-tridiagonal matrix: three-parameteric method and linear interpolation method. Some problems can be solved by these two new direct methods, while Gauss elimination method can't solve. So they are supplement to Gauss elimination method.