Using the skew-symmetric part of the coefficient matrix to find an iterative solution of the strongly nonsymmetric positive real linear system of equations (Q2734681)

Triangular iterative methods, as Seidel and successive overrelaxation (SOR) methods, are used for solving systems of linear equations. In this paper, new triangular iterative methods to solve the linear system \(Au=f\), where \(A\) is a large sparse nonsymmetric positive real matrix are presented. These methods consist of transforming the original system to the form \(B \frac{y^{n+1}-y^n}{\tau}+Ay^n=f\) or equivalently, \(Y^{n+1}=Gy^n+\tau f\), \(G=B^{-1}(B-\tau A)\), where \(\tau\) is an iterative parameter and \(B\) is an invertible matrix constructed by means of the skew-symmetric part of the matrix \(A\). NEWLINENEWLINENEWLINEWith some assumptions on the matrices \(A\) and \(B\), conditions for the convergence of the method are given depending on the parameter \(\tau\). Further, the optimal choice of this parameter is studied. NEWLINENEWLINENEWLINEFinally, the last section presents results of numerical experiments corresponding to the solution of several linear systems arising from the discretization of the steady convection-diffussion problems with the Péclet numbers \(\text{Pe}=10^3\), \(10^4\), and \(10^5\). Table~1, that should show the velocity coefficients used in the convection-diffussion equation, is missing. The number of iterations obtained using the new triangular iterative method with two different choices of matrix \(B\) and the SOR method are compared, showing that the proposed method is better in one case.