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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.

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