1. Introduction

The popular one-step Backward Euler Method (BEM) is given by the formula

(1.1)

(1.1) is known to possess a correct behaviour when the stiffness ratio is severe in a stiff system of initial value problem of ordinary differential equations given in the form

(1.2)

(1.3)

Our main concern in this paper is to propose a new family of One-Block Implicit Backward Euler Type method of the form

(1.4)

where

and

are of order k by k.

We require that the methods belonging to (1.4) converge efficiently like (1.1) but with better accuracy of order k, .

2. Derivation of Formulae (1.4)

Consider the general linear multistep method given by

(2.1)

where the step number is a variable step length, α_k and β_k are both not zero. Following Lambert (1973) by making use of Taylor expansion of(2.1) and the associated linear operator L defined as

(2.2)

where y(t) is an arbitrary function which is continuously differentiable on the interval [a, b] and expanding the test function y(t+rh) and its derivative y'(t+rh) about t gives (2.3) if we collect like terms,

(2.3)

where

and

(2.4)

For backward differentiation formulas (BDFs), we put in (2.1) and

This gives

(2.5)

If continuous coefficients of (2.5) are assumed, we have

(2.6)

where the coefficients , in formula (2.6), are evaluated at by imposing order k on it, that is . This leads to the following k by k vandermonde matrix (see Brugnano and Trigiante (1998)).

(2.7)

Equation (2.7) is used to derive k-step BDFs at point, additional methods are obtained by evaluating the derivative function at the points. This is done by computing the vandermonde matrix at . The k-step Backward Differentiation Formula and the additional methods are then combined to obtain a self-starting block that can simultaneously generate the solutions of (1.2)-(1.3) at points .

We then express the block methods for a fixed value of k like the one-step Backward Euler Method (BEM) of the form (1.4) where for the first six members are as given in this work. The computer finds it easier to compute the inverse of the vandermonde matrix with real constants in (2.7) than the matrix inverse associated with the direct construction of interpolation and collocation matrix

(2.8)

such that CD = I_kxk identity matrix. The columns of C contains the constant coefficients of the continuous multistep method. (see Onumanyi et al. (2001), Akinfenwa et al (2011)).

The first six members of the family expressed in form (1.4) have the following as

Case k = 1

Case k = 2

Case k = 3

Case k = 4

Case k = 5

Case k = 6

3. Linear Stability Analysis of 1.4

Applying (1.4) 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).

(3.6)

Case k = 1

Case k = 2

Case k = 3

Case k = 4

Case k = 5

Case k = 6

Case k = 7

Case k =8

Case k = 9

Case k = 10

Case k = 11

The characteristics polynomial of the method is

The methods are absolutely stable in the region where . Using the boundary locus method, the regions of instabilities are as given below

Figure 1. Boundary loci of the proposed blocked methods of order k, k = 1, 2, 3, …, 12

Let k denote the step number, p = k the order and

The methods are since its region of absolute stability (RAS) contains an infinite wedgeand since (see Chartier (1994))

4. Numerical Examples

Example 4.1

The test equation , with solution is solved with h = 0.01 and using different order of the proposed methods. The results of the absolute errors are displayed in table 1 below

Table 1. Absolute errors in example 4.1 when solved with different order of the proposed methods

Example 4.2

Consider the following linear constant coefficient initial value problem taken from Lambert (1973),

Its theoretical solution is

The above problem was solved by Brugnano and Trigiante (1998) using Generalized Backward Differentiation Formula (GBDF) of different order. GBDF is a boundary value method (BVM). We report here in table 2 the maximum absolute error obtained using our proposed methods and GBDF of the same order (7 and 8).

Table 2. Absolute errors in example 4.2 using the two methods

Example 4.3

The Vander Pol problem is one of the problems used to demonstrate the ability of a method to solve stiff nonlinear problems. The above Vander Pol is solved for , using order 8 of our method and step size h = 0.005. The phase diagram of the solution computed using ode 15s is displayed in figure 1 while the one by our proposed method is in figure 2 below:

Figure 2. Solution of problem 4.3 in phase plane, computed with order 8 of the proposed method ()

Figure 3. Solution of problem 4.3 in phase plane, computed with order 8 of the proposed method ( and h = 0.005)

5. Conclusions

A family of efficient high order L(α)-Stable block methods for solving stiff ordinary differential equations has been proposed and implemented for k = 1, 2, …, 12 as self-starting methods. The methods which are derived from Backward Differentiation Formula is based on order definition. It circumvented the collocation and integrand approximation approaches which have been widely used in the past by many authors. The numerical experiments show that our methods have high order and can compete favourably with existing ones.