Block Procedure with Implicit Sixth Order Linear Multistep Method using Legendre Polynomials for Solving Stiff First Order Initial Value Problems for ODEs.

Abstract

In this study, discrete sixth order implicit linear multistep methods (LMM) in block form of uniform step size for the solution of initial value problems (IVPs) for ordinary differential equations (ODEs) was presented using the Legendre polynomials. The method is based on collocation of the differential equation and interpolation of the approximate solution of power series at the grid points. The procedure yields four consistent linear multistep schemes which are combined as simultaneous numerical integrators to form block method. The method is found to be consistent and zerostable hence convergent. The accuracy of the method is tested with some standard stiff first order initial value problems. The results are compared with fourth order Runge-Kutta and with the implicit backward difference methods 2BBDF and 2BEBDF. All numerical examples show that our proposed method has a better performance over the existing methods.