A class of iteration methods based on the generalized preconditioned Hermitian and skew-Hermitian splitting for weakly nonlinear systems

For large sparse systems of weakly nonlinear equations, based on the separable property and strong dominance between the linear and the nonlinear terms, Bai and Yang studied two nonlinear composite iteration schemes called Picard-HSS and nonlinear HSS-like methods (see [Z.-Z. Bai, X. Yang, On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math. 59 (12) (2009) 2923–2936]). In this paper, we generalize these methods and propose a class of generalized nonlinear composite splitting iteration schemes called Picard-GPHSS and nonlinear GPHSS-like iteration methods, to solve the large sparse systems of weakly nonlinear equations. We derive conditions for guaranteeing the local convergence of these iterative methods and derive some special case of iterative methods by choosing different parameters and preconditioned matrices. Numerical experiments are used to demonstrate the feasibility and effectiveness. And an efficient preconditioner is presented for the new methods in actual implementation. The efficiency is effectively testified by some comparisons with numerical results.