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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Diagonally Implicit Super Class of Block Backward Differentiation Formula with Off‐Step Points for Solving Stiff Initial Value Problems

Babangida B* and Musa H

Department of Mathematics and Computer Sciences, Umaru Musa Yar'adua University, Katsina, Nigeria

*Corresponding Author:
Babangida B
Department of Mathematics and Computer Sciences
Faculty of Natural and Applied Sciences
Umaru Musa Yar'adua University
Katsina, Nigeria
Tel: +2347067704150
E-mail: bature.babangida@umyu.ed.ng

Received Date: September 13, 2016; Accepted Date: September 27, 2016; Published Date: October 03, 2016

Citation: Babangida B, Musa H (2016) Diagonally Implicit Super Class of Block Backward Differentiation Formula with Off-Step Points for Solving Stiff Initial Value Problems. J Appl Computat Math 5: 324. doi: 10.4172/2168-9679.1000324

Copyright: © 2016 Babangida B, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

A new formula called 2-point diagonally implicit super class of BBDF with two off-step points (2ODISBBDF) for solving stiff IVPs is formulated. The method approximates two solutions with two off-step points simultaneously at each iteration. By varying a parameter ρ ∈ (–1,1) in the formula, different sets of formulae can be generated. A specific choice of ρ =3/4  is made and it was shown that the method is both zero and A-stable. A comparison between the new method and the existing 2-point block backward differentiation formula with off-step points (2OBBDF) is made. The results show that the new method outperformed existing 2OBBDF method in terms of accuracy.

Keywords

Off-step; Diagonally implicit super class of block backward differentiation formula; Stiff IVPs; Implicit block method; A-stability

Introduction

Consider a system of first order stiff initial value problems (IVPs) of the form:

image

With image in the interval a ≤ x ≤ b where image and image.

System (1) is said to be stiff if it contains widely varying time scales, i.e., some components of the solution decay much more rapidly than others. Most realistic stiff systems do not have analytical solutions so that a numerical procedure must be used. Stiff ODEs occur in many fields of engineering and physical sciences such as electrical circuits, vibrations, chemical reactions, kinetics etc.

Developing methods for solving stiff problems remains a challenge in modern numerical analysis. Curtiss and Hirschfelder [1] discover Backward Differentiation Formula (BDF). Since then most of the improvements in the class of linear multistep methods have been based on BDF because of its special properties. Ibrahim [2] introduced r-point block BDF (BBDF). Super class of block BDF, which is both zero and A-stable, was developed by Suleiman [3,4]. The method is derived from 2-point block BDF and outperformed both 2BBDF and 1BDF in terms of accuracy.

In order to gain an efficient numerical approximation in terms of accuracy and computational time, a super class of diagonally implicit BBDF method can be considered. The study of diagonally implicit for multistep attracted some researchers such Ababneh [5], Alexander [6], Musa [7] and Zawawi [8]. Abasi [4] developed a 2-point Block BDF Method with off-step points for solving Stiff ODEs which differs from all the methods above because it calculates two solution values with off-step points simultaneously at each iteration. The motivation of this research is to develop a new method that would be called diagonally implicit superclass of BBDF with off-step points.

Derivation

In this work, two solution values, yn+1 and yn+2 with step size h, and two off-step points image and image which are chosen at the points where the step size is halved, are formulated in a block simultaneously. The formulae are computed using two back values yn-1 and yn with step size h. The formula is derived with the aid of this diagram below (Figure 1).

applied-computational-mathematics-points

Figure 1: Points involved in 2-point super class BBDF with off-step points method.

Definition 2.1: The 2-point diagonally implicit super class of block backward differentiation formula with off-step points is defined as:

image

Where, image represents the first point, k = i = 1 represents the second point, image represents the third point and k = i = 2 represents the fourth point. The formula (2) is derived from Taylor’s series expansion as follows:

Definition 2.2: Linear operator Li associated with first, second, third and fourth point DI2SBBDF with off-step point is defined by:

image

image

image

image

First point: To derive the first point image let image and define the operator as

image

Expanding (7) as Taylor series about xn and collecting like terms gives

image

Where,

image

image

image

The coefficient of image is normalized to 1. Solving the simultaneously equation (9) for the values of image and image gives the formula for image as

image

Similar procedure is applied as in the derivation of first point to obtain the second, third and fourth points as:

image

image

image

For absolute stability of the method, ρ is Chosen to be in the interval (–1,1) as in Suleiman [3]. By choosing image in equation (10), (11), (12) and (13) to obtain the 2–point diagonally implicit super class of BBDF with off-step points as follows:

image

image

image

image

In matrix form, equation (14) can be written as

image

image

Definition 2.3: Method (15) is diagonally implicit if the matrix in its left hand side is an upper triangular.

Order of the Method

This section derives the order of the method corresponding to the equations in (14). It can be written in the following form:

image

image

image

image

Equation (16) can be written as in matrix form as:

image

image

image

image

Definition 3.1: The order of the block method (14) and its associated linear operator L given by:

image

Expanding the function image and its derivative image as Taylor series around x gives

image

image

Substituting (19) and (20) into (18) represents

image

image

image

image

The difference operator (21) and the associated method (14) is considered of order p if E0 = E1 = E2 = … = Ep = 0 and Ep+1 ≠ 0

In this case

image

image

image

image

Therefore, the method (14) is of order 2, with error constant

image

Stability Analysis of SBBDF with Off-Step Point

The stability properties of method (14) are discussed here. We begin by defining zero and A-stability taken from Suleiman [3].

Definition 4.1: A LMM is said to be zero stable if no root of the first characteristics polynomial has modulus greater than one and that any root with modulus one is simple.

Definition 4.2: A LMM is said to be A-stable if its stability region covers the entire negative half-plane

The method (14) can be rewritten in matrix form as follows:

image

image

Definition 4.3: Let Ym and Fm be vectors defined by

Ym = [yn+1, yn+1,…, yn+r]T, Fm = [fn+1,fn+2,…,fn+r]T r = 2, and n = 2m (See Suleiman [3]).

Method (14) can be written in matrix form as follows:

A0Ym = A1Ym–1+h(B0Fm–1+B1Fm). (27)

Where image

image

Substituting scalar test equation y′ = λy(λ<0, λ complex) into (27) and using image gives

image

The stability polynomial of (14) is given by

image

i.e.,

image

For zero stability, we set image in (30) to obtain:

image

Solving equation (31) for t gives the following roots:

t = 0, t = 0, t = 0.350014 and t = 1. (32)

From the definition 4.1, method (14) is zero-stable. The stability region of method (14) is determined by substituting t = e and the graph is shown below (Figure 2):

applied-computational-mathematics-region

Figure 2: Stability region of the 2-point super class BBDF with off-step points.

From the definition 4.2, method (14) is A-stable.

Implementation of the Method

Newton’s iteration is used in implementing the method. The procedure is described as follows. We begin by defining the error.

Definition 5.1

Let yi and y(xi) be the approximate solution of (1)

Then the absolute error is given by:

(errori)t = |(yi)t –(y(xi))t|. (33)

The maximum error is defined by

image

Where, T is the total number of steps and N is the number of equations.

Define from (14)

image

image

image

image

Where image and image are the back values.

Let image denote the (i + 1)th iterative values of yn+j and define

image

Newton’s iteration for the 2-point SBBDF with off-step point method takes the form:

image

This can be written as

image

and in matrix form, equation (38) is equivalent to

image

image

Tested Problems

To validate the efficiency of the methods developed, the following stiff IVPs are solved:

image

Exact solutions: y1(x) = e–39x+ e–x

y2(x) = e–39x+ e–x

Eigen values: -1 and -39

Source: Musa [9]

image

Exact solution: y1(x) = e–x

y2(x) = e–x

Eigen values: -1 and -200

Source: Ibrahim [2].

image

Exact Solution: y(x) = sin x + e–20x.

Source: Abasi [4]

image

Exact Solution: y(x) = e–100x + x

Source: Abasi [4]

Numerical Results

The numerical results for the test problems given in section 6 are tabulated. The problems are solved with 2OBBDF and 2ODISBBDF methods. The number of step taken to complete integration and maximum error for the different methods is presented and compared in the tables below. In addition, the graph of Log10 (MAXE) against h for each problem is plotted (Figures 2-6) in order to give the visual impact on the performance of the method. The notations used in the tables are listed below:

applied-computational-mathematics-problem

Figure 3: Graph of Log10 (MAXE) against h for problem 1.

applied-computational-mathematics-graph

Figure 4: Graph of Log10 (MAXE) against h for problem 2.

applied-computational-mathematics-log

Figure 5: Graph of Log10 (MAXE) against h for problem 3.

applied-computational-mathematics-maxe

Figure 6: Graph of Log10 (MAXE) against h for problem 4.

The following abbreviations are used in the tables:

2ODISBBDF = 2-point super class BBDF with off-step points

2OBBDF = 2-point block BDF method with off-step points of order 5

h = step size

NS = total number of steps

MAXE = maximum error

Time = computational time in seconds (Table 1).

  H Method NS MAXE Time
1. 10–2
10–4
10–6
2OBBDF
2ODISBBDF
2OBBDF
2ODISBBDF
2OBBDF
2ODISBBDF
1000
1000
100000
100000
10000000
10000000
7.00088e-002
3.81561e-002
2.84492e-003
1.64714e-005
2.87417e-005
1.70657e-009
2.15117e-003
1.73800e-001
2.06491e-001
2.01139e+000
6.00132e+001
1.19700e+002
2. 10–2
10–4
10–6
2OBBDF
2ODISBBDF 2OBBDF
2ODISBBDF
2OBBDF
2ODISBBDF
500
500
50000
50000
5000000
5000000
7.17251e-003
1.03577e-004
7.35564e-005
1.12034e-008
7.35775e-007
1.96752e-010
1.40432e-003
1.68700e-001
1.41352e-001
1.13470e+000
2.41100e+000
9.01800e+001
3. 10–2
10–4
10–6
2OBBDF
2ODISBBDF 2OBBDF
2ODISBBDF
2OBBDF
2ODISBBDF
100
100
10000
10000
100000
100000
8.05923e-002
1.86882e-002
1.46355e-003
4.39784e-006
1.47126e-005
4.48628e-010
5.90201e-004
1.18580e-001
2.01000e-002
5.03090e-001
2.98923e+000
3.76200e+001
4. 10–2
10–4
10–6
2OBBDF
2ODISBBDF 2OBBDF
2ODISBBDF
2OBBDF
2ODISBBDF
500
500
50000
50000
5000000
5000000
1.95750e-002
2.62911e-002
7.16455e-003
1.03577e-004
7.35564e-005
1.12034e-008
3.08300e-003
1.99800e-001
5.92900e-002
1.38300e+000
9.91000e+000
1.15600e+002

Table 1: Numerical results for problems 1, 2, 3 and 4.

To give the visual impact on the performance of the method, the graphs of Graph of Log10 (MAXE) against h for the problems tested are plotted. Given below are the graphs of the scaled maximum error arranged problem by problem.

From the table 1 above it can be seen that 2ODISBBDF method outperformed 2OBBDF method in terms of accuracy. The graphs also show that the scaled errors for the 2ODISBBDF method are smaller when compared with that in 2OBBDF method.

Conclusion

A new method of order 2 that is suitable for solving stiff initial value problems has been developed. The stability analysis has shown that the method is both zero and A-stable. Accuracy and the execution time of the derived method are compared with the existing 2-point block backward differentiation formula with off-step points (2OBBDF). This comparison shows that the new method outperformed the existing 2OBBDF method in terms of accuracy. The computation time for the new method is seen to be competitive. The graphs also show that the scaled errors for the 2ODISBBDF method are smaller when compared with that in 2OBBDF method.

References

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