On asymmetric Hermitian and skew-Hermitian splitting iteration methods for weakly nonlinear systems

For the large sparse system of weakly nonlinear equations, based on separable property and strong dominance between the linear and the nonlinear terms and on the asymmetric Hermitian and skew-Hermitian splitting (AHSS) of the coefficient matrix, we establish two nonlinear composite iteration schemes, called Picard-AHSS and nonlinear AHSS-like methods. The advantage of these methods is that they only need to solve the linear sub-systems of constant coefficient matrices. In addition, these methods can also take full advantage of the dominance of Hermitian part. Therefore, computational workloads and computer storage may be saved in actual implementations. Theoretical analysis show that these new iteration methods are local convergent under suitable conditions. Numerical results show that both Picard-AHSS and nonlinear AHSS-like iteration methods are feasible and effective for the large scale system of weakly nonlinear equations, especially when the Hermitian part of the coefficient matrix is dominant.