American Journal of Computational and Applied Mathematics

p-ISSN: 2165-8935    e-ISSN: 2165-8943

2014;  4(1): 24-31

doi:10.5923/j.ajcam.20140401.04

A Family of L(α) − stable Block Methods for Stiff Ordinary Differential Equations

Ajie I. J.1, Ikhile M. N. O.2, Onumanyi P.1

1National Mathematical Centre, Abuja, Nigeria

2Department of Mathematics, University of Benin, Benin City, Nigeria

Correspondence to: Ajie I. J., National Mathematical Centre, Abuja, Nigeria.

Email:

Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.

Abstract

A family of self-starting L(α)-stable linear multistep block methods (LMBMs) for solving stiff initial value problems (IVPs) in ordinary differential equations (ODEs) is proposed. The constructions are done by pairing three-step Top Order Method (TOM) and k-step Backward Differentiation Formulas (BDF) and using the shifting techniques introduced by Ajie et al (2011). The performance of the resultant block methods on the numerical examples considered shows their effectiveness.

Keywords: Top Order Method (TOM), Backward Differentiation Formulas (BDF), L(α) −stable, A(α) −stable,Linear Multistep Block Methods (LMBMs)

Cite this paper: Ajie I. J., Ikhile M. N. O., Onumanyi P., A Family of L(α) − stable Block Methods for Stiff Ordinary Differential Equations, American Journal of Computational and Applied Mathematics , Vol. 4 No. 1, 2014, pp. 24-31. doi: 10.5923/j.ajcam.20140401.04.

Article Outline

1. Introduction
2. Construction of the Methods
3. Stability Analysis of the Methods (See Akinfenwa et al. (2011))
4. Numerical Experiments
5. Conclusions

1. Introduction

Many physical problems when modeled mathematically result in ordinary differential equations. Some of these equations don’t have solution in closed form. Therefore there is need to provide good numerical methods to approximate their solutions. In this paper, we introduced aself-starting L(α)-stable family of order 4, 5 and 6 for stiff initial value problem of the form
(1.1)
It is assumed that f is continuous and satisfies Lipschitz condition as specified in Henrici (1962). Solving (1.1) using k-step linear multistep methods require
The provision of k-1 starting values
A-stable method if it is stiff
High order method if high accuracy is needed.
Standard linear multistep methods suffer from Dalquist order barrier theorem that no A-stable method can have order greater than two. Many steps have been taken by different numerical analyst to circumvent this barrier and construct high order A-stable schemes as can be seen in Brugnano and Trigiante (1998), Akinfenwa et al (2011), Ajieet al (2011).
Curtiss and Hirshfelder (1952) introduced backward differentiation formulas (BDFs) also called Gear’s methods (because it was used extensively by Gear). Cases for k ≤ 5 played important role in many algorithms for solving stiff systems by initial value methods (IVMs).
Block methods were also introduced to both circumvent the barrier and provide the k-1 starting values to k-step LMM. They have the capacity to generate simultaneously k approximate solutions. For block methods see the following references Akinfenwa (2011), Ajie et al (2011), Fatunla (1991, 1994), Zanariah and Suleiman (2007), Cash and Diamantakis (1994), Brugnano and Magherini (2000, 2007), Muka and Ikhile (2012), Brugnano and Trigiante (1997, 1998), Shampine and Watts (1972), Onumanyiet al. (1994, 2001), Bond and Cash (1979).
In the section that follows, we give the construction of the methods; section three gives the stability analysis. In section four, we give numerical examples and conclude in section five.

2. Construction of the Methods

The general linear multistep formula (LMF) is given by
(2.1)
where is the variable step size, , are coefficients which can be uniquely determined.
By introducing two polynomials , (2.1) can be rewritten as
(2.2)
Taking in its simplest form, we obtain a class of order k,k-step LMF known as BDF and are given by
(2.3)
A popular example is the implicit Euler method, . which is the only L-stable member of the BDF family known of order one. It is in fact known that BDF of order are zero unstable hence will not converge. Cases of k=4, 5 and 6 will be used in our constructions.
Almost all the families of linear multistep methods are obtained by imposing one restriction or the other on the structure of the polynomials or . The family constructed without ‘a priori’ restrictions on such polynomials by imposing maximum order on (2.1) in order to get the coefficients is called TOMs. This familywas considered by Dahlquist (1956) and presented as unstable methods. Exampleis the Sixth Order TOM below which we will use in constructing our family of methods.
(2.4)
Their stability properties as boundary value methods (BVMs) were studied by Amodio (1996).
The objective of this paper is to derive efficient higher order block methods which are L(α) − stable among block LMF for efficiency. Higher order methods are considered for where p is theorder. These are obtained by pairing TOMs with k = 3, order p = 2k and the BDF with each of order k.The pairs are then shifted forwards imultaneously repeatedly timesto give a set of 2(k - 1) equations which can be solved to obtain the unknowns . For values of , the construction yields the new one step block method of the form
(2.5)
where
(2.6)
Case k = 4
Case k = 5
Case k = 6
We have constructed the one-step block LMF of order in this section using only a pair of LMF. The problem of looking for additional methods to form blocks are overcome.

3. Stability Analysis of the Methods (See Akinfenwa et al. (2011))

Applying (2.5) to the test equation
(3.1)
gives
(3.2)
where
(3.3)
From (3.3), we obtain the stability function R(z) which is a rational function of real coefficients given by
(3.4)
where I is a identity matrix. The stability domain S of a one-step block method is
(3.5)
From the analysis given we easily obtain as follows the rational functions R(z) for which satisfies (3.5).
k = 4
k = 5
k = 6
The boundary locus plots of the stability domain for k= 4, 5, and 6 are given as follows:
This is said to be A-stable (A(90o)-stable).
This is said to be A(82o)-stable.
This is said to be A(81o)-stable.
Hence we conclude the stability analysis of the three new methods in section 3.0 to be
since . (see Chartier (1993))

4. Numerical Experiments

Example 4.1
The test equation is solved with h = 0.01 using the three methods, order 4, 5 and 6 and the results displayed in the table below
It can easily be seen that the accuracy increases as the order increases.
Example 4.2 Lorenz chaotic attractor (see Sparrow (1982))
; ;
The above Lorenz equation is solved using order 6 of our method and step size h = 0.01. The result is compared with the one computed using ode45 and are displayed in the figure below
The blue dot is the computed results using our new block method while the red line is the one computed with ode45.
Example 4.3
The Vander Pol problem is used to demonstrate the ability of the methods to solve stiff nonlinear problems. The above Vander Pol is solved for , using order 6 of our method and step size h = 0.01. The result is displayed in the figure below

5. Conclusions

A new family of block methods that is self-starting has been introduced. They combine the accuracy of TOMs and the stability properties of BDFs in the block implicit pairs formed for adequate block-by-block forward integration. The analysis of the stability properties shows that the methods are for . The numerical experiments considered showed that the methods are comparable to existing ones.

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