Higher order multi-step iterative method for computing the numerical solution of systems of nonlinear equations: Application to nonlinear PDEs and ODEs

摘要

In the present study, we consider multi-step iterative method to solve systems of nonlinear equations. Since the Jacobian evaluation and its inversion are expensive, in order to achieve a better computational efficiency, we compute Jacobian and its inverse only once in a single cycle of the proposed multi-step iterative method. Actually the involved systems of linear equations are solved by employing the LU-decomposition, rather than inversion. The primitive iterative method (termed base method) has convergence-order (CO) five and then we describe a matrix polynomial of degree two to design a multi-step method. Each inclusion of single step in the base method will increase the convergence-order by three. The general expression for CO is 3s - 1, where s is the number of steps of the multi-step iterative method. Computational efficiency is also addressed in comparison with other existing methods. The claimed convergence-rates proofs are established. The new contribution in this article relies essentially in the increment of CO by three for each added step, with a comparable computational cost in comparison with existing multi-steps iterative methods. Numerical assessments are made which justify the theoretical results: in particular, some systems of nonlinear equations associated with the numerical approximation of partial differential equations (PDEs) and ordinary differential equations (ODEs) are built up and solved. (C) 2015 Elsevier Inc. All rights reserved.