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The space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix and a symmetric positive definite diagonal-plus-Toeplitz matrix. The Hermitian and skew-Hermitian splitting (HSS) method and the partially inexact HSS (PIHSS) method are employed to solve the discretized linear system. In the inner iteration processes of the HSS method, we only need to solve the linear sub-systems associated with the Hermitian part inexactly by the conjugate gradient (CG) method, resulting in PIHSS iteration method. Theoretical analyses show that both HSS and PIHSS methods are unconditionally convergent. Numerical examples are given to demonstrate the effectiveness of the HSS iteration and the PIHSS iteration. [ABSTRACT FROM AUTHOR]

The implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable, is employed to discretize the space fractional coupled nonlinear Schrödinger equations. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric diagonal-plus-Toeplitz matrix. In this paper, the HSS-like iteration method is proposed to solve the discretized linear system. Theoretical analyses show that the HSS-like iteration method is unconditionally convergent. Numerical examples are presented to illustrate the effectiveness of the HSS-like iteration method. [ABSTRACT FROM AUTHOR]

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050–3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver. [ABSTRACT FROM AUTHOR]

Abstract: In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, we present block SSOR and modified block SSOR iteration methods based on the special structures of the coefficient matrices. In each step of the block SSOR iteration, we employ the block LU factorization to solve the sub-systems of linear equations. We show that the block LU factorization is existent and stable when the coefficient matrices are block diagonally dominant of type-II by columns. Under suitable conditions, we establish convergence theorems for both block SSOR and modified block SSOR iteration methods. In addition, the block SSOR iteration and AF-ADI method are considered as preconditioners for the nonsymmetric systems of linear equations. Numerical experiments show that both block SSOR and modified block SSOR iterations are feasible iterative solvers and they are also effective for preconditioning Krylov subspace methods such as GMRES and BiCGSTAB when used to solve this class of systems of linear equations. [Copyright &y& Elsevier]