Modified Newton Preconditioned Block GADI Method for Complex Nonlinear Systems

Abstract

An efficient modified Newton-PBGADI method for nonlinear systems with large sparse non-Hermitian positive definite Jacobian matrices is proposed. The method integrates a modified Newton framework that achieves $R-$order three convergence while requiring only a single inversion of the Jacobian matrix per iteration. The resulting linear subproblems are solved by a preconditioned block GADI method. The convergence of this iterative scheme is studied and the local convergence of the overall modified Newton–PBGADI method is rigorously analyzed. Two numerical examples involving two-dimensional nonlinear PDEs validate effectiveness and computational efficiency of the method.