Block procedure with implicit sixth order linear multistep method using Legendre polynomials for solving stiff initial value problems

Abstract

In this paper, a discrete implicit linear multistep method in block form of uniform step size for the solution of first-order ordinary differential equations is presented using the power series as a basis function. To improve the accuracy of the method, a perturbation term is added to the approximated solution. The method is based on collocation of the differential equation and interpolation of the approximate solution using power series at the grid points. The procedure yields four linear multistep schemes which are combined as simultaneous numerical integrators to form block method. The method is found to be consistent and zero-stable, and hence convergent. The accuracy of the method is tested with some standard stiff first order initial value problems. All numerical examples show that our proposed method has a better accuracy than some existing numerical methods reported in the literature.