Derivation of Diagonally Implicit Block Backward Differentiation Formulas for Solving Stiff Initial Value Problems

Iskandar Shah Mohd Zawawi,Zarina Bibi Ibrahim,Khairil Iskandar Othman

doi:10.1155/2015/179231

Abstract

The diagonally implicit 2-point block backward differentiation formulas (DI2BBDF) of order two, order three, and order four are derived for solving stiff initial value problems (IVPs). The stability properties of the derived methods are investigated. The implementation of the method using Newton iteration is also discussed. The performance of the proposed methods in terms of maximum error and computational time is compared with the fully implicit block backward differentiation formulas (FIBBDF) and fully implicit block extended backward differentiation formulas (FIBEBDF). The numerical results show that the proposed method outperformed both existing methods.

Highlights

Many scientific and engineering problems which arise in real-life applications are in the form of ordinary differential equations (ODEs), where the analytic solution is unknown
In the early 1950s, Curtiss and Hirschfelder [1] realized that there is an important class of ODEs which is known as stiff initial value problems (IVPs)
The performance of the derived methods is compared with the existing methods in terms of maximum error and exe- 10−2 cution time

Summary

Introduction

Many scientific and engineering problems which arise in real-life applications are in the form of ordinary differential equations (ODEs), where the analytic solution is unknown. The general form of first order ODEs is given in the following form: y󸀠 = f (x, y) , y (a) = y0, a ≤ x ≤ b. In the early 1950s, Curtiss and Hirschfelder [1] realized that there is an important class of ODEs which is known as stiff initial value problems (IVPs). There are various definitions of stiffness given in the literature. Stiff problems are problems where certain implicit methods perform better than explicit ones. We choose the definition of stiff problem given by Lambert [2]. The system of (1) is said to be stiff if (1) Re(λt) < 0, t = 1, 2, .

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