Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs

Yohanna Sani Awari

doi:10.11648/j.sjams.20170505.13

Abstract

A Modified Three Step Block Hybrid Extended Trapezoidal Multistep Method of Second Kind (BHETR<sub>2</sub>s) with two off-grid points, one at interpolation and another at collocation point yielding uniform order six (6, 6, 6, 6, 6)<sup>T</sup> for the Numerical Integration of initial value problems of stiff Ordinary Differential Equations was developed. The main method and additional equations were obtained from the same continuous formulation through interpolation and collocation procedures. The stability properties of the method was discussed and from the stability region obtained, the method is suitable for the solution Stiff Ordinary Differential Equations. Three numerical examples were considered to illustrate the efficiency and accuracy.

Highlights

Consider the stiff initial value problem in the form: y′(x) = f (x, y), y(a) = y0 (1)on the finite interval I = x0, xN, where y : [x0, xN ] → Rm and f : [x0, xN ]× Rm → Rm are continuous
A more elegant and computationally attractive procedurewas proposed in this paper, which leads to a class of stable general linear methods for stiff systems of initial value problems
Evaluating the continuous formulation in (3) yields the BHETR2sassociated with the continuous scheme and converting it into A, B,U and V of the General Linear Method (12) as:

Summary

Introduction

Consider the stiff initial value problem in the form: y′(x) = f (x, y), y(a) = y0. To approximate the solution of problem (1) over the considered partition. The numerical solution of linear and nonlinear system of stiff system can be found in [3, 11 and 12]. Block methods were introduced to both improve the stability of methods and provide the k − 1 starting values to k − step LMM. They can be seen as a set of linear multistep methods simultaneously applied to (1) and combined to yield better approximations (Ajie, et al, 2014)

Formulation of the Method

Order of BHETR2s

Absolute Stability Region

Convergence

Implementation Strategies

Conclusions