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An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs. (English) Zbl 1328.65156
Summary: We developed multi-step iterative method for computing the numerical solution of nonlinear systems, associated with ordinary differential equations (ODEs) of the form \(L(x(t)) + f(x(t)) = g(t)\): here \(L(\cdot)\) is a linear differential operator and \(f(\cdot)\) is a nonlinear smooth function. The proposed iterative scheme only requires one inversion of Jacobian which is computationally very efficient if either LU-decomposition or GMRES-type methods are employed. The higher-order Frechet derivatives of the nonlinear system stemming from the considered ODEs are diagonal matrices. We used the higher-order Frechet derivatives to enhance the convergence-order of the iterative schemes proposed in this note and indeed the use of a multi-step method dramatically increases the convergence-order. The second-order Frechet derivative is used in the first step of an iterative technique which produced third-order convergence. In a second step we constructed matrix polynomial to enhance the convergence-order by three. Finally, we freeze the product of a matrix polynomial by the Jacobian inverse to generate the multi-step method. Each additional step will increase the convergence-order by three, with minimal computational effort. The convergence-order (CO) obeys the formula \({\mathrm{CO}} = 3 m\), where \(m\) is the number of steps per full-cycle of the considered iterative scheme. Few numerical experiments and conclusive remarks end the paper.

MSC:
65L05 Numerical methods for initial value problems
34A45 Theoretical approximation of solutions to ordinary differential equations
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