Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: Application to PDEs and ODEs

This paper proposes a multi-step iterative method for solving systems of nonlinear equations with a local convergence order of 3m−4, where m(≥2) is the number of steps. The multi-step iterative method includes two parts: the base method and the multi-step part. The base method involves two function evaluations, two Jacobian evaluations, one LU decomposition of a Jacobian, and two matrix-vector multiplications. Every stage of the multi-step part involves the solution of two triangular linear systems and one matrix-vector multiplication. The computational efficiency of the new method is better than those of previously proposed methods. The method is applied to several nonlinear problems resulting from discretizing nonlinear ordinary differential equations and nonlinear partial differential equations.