A higher order multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with nonlinear PDEs and ODEs

The main focus of research in the current article is to address the construction of an efficient higher order multi-step iterative methods to solve systems of nonlinear equations associated with nonlinear partial differential equations (PDEs) and ordinary differential equations (ODEs). The construction includes second order Frechet derivatives. The proposed multi-step iterative method uses two Jacobian evaluations at different points and requires only one inversion (in the sense of LU-factorization) of Jacobian. The enhancement of convergence-order (CO) is hidden in the formation of matrix polynomial. The cost of matrix vector multiplication is expensive computationally. We developed a matrix polynomial of degree two for base method and degree one to perform multi-steps so we need just one matrix vector multiplication to perform each further step. The base method has convergence order four and each additional step enhance the CO by three. The general formula for CO is 3s - 2 for s ≥ 2 and 2 for s = 1 where s is the step number. The number of function evaluations including Jacobian are s + 2 and number of matrix vectors multiplications are s. For s-step iterative method we solve s upper and lower triangular systems when right hand side is a vector and 1 pair of triangular systems when right hand side is a matrix. It is shown that the computational cost is almost same for Jacobian and second order Frechet derivative associated with systems of nonlinear equations due to PDEs and ODEs. The accuracy and validity of proposed multi-step iterative method is checked with different PDEs and ODEs.