Nonsymmetric preconditioner updates in Newton-Krylov methods for nonlinear systems

Newton–Krylov methods, a combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners, and this step is very often the dominant cost of Newton–Krylov methods. Therefore, working with preconditioners may destroy the “Jacobian-free” (or matrix-free) setting where the single Jacobian-vector product can be provided without forming and storing the element of the true Jacobian. In this paper, we propose and analyze a preconditioning technique for sequences of nonsymmetric Jacobian matrices based on the update of an earlier preconditioner. The proposed strategy can be implemented in a matrix-free manner. Numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with the standard ILU-preconditioned Newton–Krylov approaches.