On HSS-based iteration methods for weakly nonlinear systems

Based on separable property of the linear and the nonlinear terms and on the Hermitian and skew-Hermitian splitting of the coefficient matrix, we present the Picard-HSS and the nonlinear HSS-like iteration methods for solving a class of large scale systems of weakly nonlinear equations. The advantage of these methods over the Newton and the Newton-HSS iteration methods is that they do not require explicit construction and accurate computation of the Jacobian matrix, and only need to solve linear sub-systems of constant coefficient matrices. Hence, computational workloads and computer memory may be saved in actual implementations. Under suitable conditions, we establish local convergence theorems for both Picard-HSS and nonlinear HSS-like iteration methods. Numerical implementations show that both Picard-HSS and nonlinear HSS-like iteration methods are feasible, effective, and robust nonlinear solvers for this class of large scale systems of weakly nonlinear equations.