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

- *Corresponding Author:
- B. Babangida

Department of Mathematics and Computer Sciences

Faculty of Natural and Applied Sciences

Umaru Musa Yar'adua University Katsina

Katsina State, Nigeria

**Tel:**+2347067704150

**E-mail:**[email protected]

**Received Date**: September 25, 2016; **Accepted Date:** October 23, 2016; **Published Date**: October 30, 2016

**Citation: **Babangida B, Musa H, Ibrahim L. K. (2016) A New Numerical Method for
Solving Stiff Initial Value Problems. Fluid Mech Open Acc 3: 136.

**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.

**Visit for more related articles at** Fluid Mechanics: Open Access

A new numerical method that computes 2–points simultaneously at each step of integration is derived. The numerical scheme is achieved by modifying an existing DI2BBDF method. The method is of order 2. The stability analysis of the new method indicates that it is both zero and A–stable, implying that it is suitable for stiff problems. The necessary and sufficient conditions for the convergence of the method are also established which proved the convergence of the method. Numerical results show that the method outperformed some existing algorithms in terms of accuracy.

A-Stability; Order of a block method; Implicit block method; Stiff initial value problems; Convergence; Diagonally implicit; Zero stability

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

(1)

System (1) can be regarded as stiff if its exact solution contains very fast as well as very slow components [1]. Stiff IVPs occur in any fields of engineering and physical sciences. They are particularly found in electrical circuits, vibrations, chemical reactions, kinetics, automatic control and combustion, theory of fluid mechanics etc. The solution is characterized by the presence of transient and steady state components, which restrict the step size of many numerical methods except methods with A-stability properties (Suleiman [2,3]). This behaviour makes it difficult to develop suitable methods for solving stiff problems. However, efforts have been made by researchers, such as Abasi [4], Alt [5], Alvarez [6], Cash [7], Dahlquist [1], Ibrahim [8-10], Musa [11-14], Suleiman [2,3], Yatim [15] and Zawawi [16] among others, to develop methods for stiff ODEs. The need to obtain an efficient numerical approximation in terms of accuracy and computational time have attracted some researchers such as Alexander [17] with diagonally implicit Runge-Kutta for stiff ODEs, Ababneh [18] with design of new diagonally implicit Runge-Kutta for stiff problems, Ismail [19] with embedded pair of diagonally implicit Runge-Kutta for solving ODEs, Zawawi [20] with diagonally implicit block backward differentiation formulas for solving ODEs. The motivation of this research is to modify the method developed by Zawawi [20] so as to improve its accuracy and stability properties.

This section describes the derivation of the method. Consider the numerical scheme developed by Zawawi [20].

(2)

To improve its accuracy and stability, the term -hβ(k,i) ρf(n+k-1) is added to (2) to come up with new scheme as follows

(3)

Where, k=i=1 represents the first point, k=i=2 represents the second point and ρ ∈ (-1, 1). In this paper, the value is used. The formula (3) is derived from Taylor’s series expansion.

A Linear operator L_{i} for the first and second point of the new
method is defined by:

(4)

(5)

respectively.

Expanding (4) and (5) as Taylor’s series about x_{n}, collect like terms
and normalized the coefficient of the first point α_{2,1} and second point
α_{3,2} to obtain the following implicit 2–point block formula:

(6)

The error constant of the new method (6) is implying that is of order 2.

Throughout this paper, the method will be referred to New Diagonally Implicit Super Class of Block Backward Differentiation Formula (NDISBBDF).

This section presents the stability analysis of the method (6). It begins by presenting the definition of zero and A-stability taken from Suleiman [2].

**Definition 3.1:** A linear multistep method (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 3.2:** A linear multistep method (LMM) is said to be
A-stable if its stability region covers the entire negative half-plane.

Formula (6) can be written in matrix form as follows

(7)

Equation (7) can be rewritten in the following form:

(8)

Where,

Substituting the scalar test equation is complex) into (8) and using gives

(9)

The stability polynomial of (6) is obtained by evaluating:

(10)

to get,

(11)

To show that the method (6) is zero stable, we set in (11) to get the first characteristics polynomial as follows:

(12)

Solving equation (12) for t gives the following roots:

t=0.1159 and t=1 (13)

From the definition 3.1, method (6) is zero-stable.

The boundary of the stability region of (6) is determined by
substituting into (11). The graph of stability region for (6)
using maple is given in **Figure 1**.

The stability region covers the entire negative half plane indicating that the method (6) is A-stable.

Convergence is an essential feature that every acceptable linear multistep method (LMM) must possess. This section discussed the convergence of the method (6). Consistency and zero stability are the necessary and sufficient conditions for the convergence of any numerical scheme. In section 3, it was shown that method (6) is zero stable. It is now remain to show that method (6) is consistent.

This discussion will be based on matrix form of (6) which can be written as:

(14)

With

**Definition 4.1:** Method (6) is consistent if and only if the following
conditions are satisfied:

(15)

(16)

Where, D_{j,s} and G_{j,s} are defined above.

Equation (15) and (16) then become

(17)

(18)

Thus, the consistency conditions in (15) and (16) are therefore met. Hence, method (6) is consistent.

Since the method (6) is both consistent and zero stable, it is thus converges.

This section discussed the implementation of the method using Newton iteration and begin by defining the absolute and maximum error.

**Definition 5.1:** Let y_{i} and y(x_{i}) be the approximate and exact
solution of (1) respectively. Then the absolute error is given by

(19)

The maximum error is given by

(20)

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

Define;

Where,

Let denote the (i+1)th iteration and

(22)

(23)

This can be written in the form:

(24)

Newton’s iteration for the new method takes the form:

(25)

In addition, in matrix form, equation (25) is equivalent to

(26)

The following problems are used to test the performance of the method.

**Problem 1 (Musa [13])**

Exact solution:

**Eigenvalues:** -1 and -39.

**Problem 2**

**Exact solution:**,

**Eigenvalues:** − 2 and − 96.

**Problem 3**

**Exact solution:**

NDISBBDF=New Diagonally Implicit Super Class of Block Backward Differentiation Formula

NS=Total Number of Steps

MAXE=Maximum Error

Time=Computational Time in Seconds

h=Step Size

To give the visual impact on the performance of the new method,
the graphs of Log10 (MAXE) against h for the problems tested are
plotted in **figures 2-4**.

From **tables 1-3**, it can be seen that the new method outperformed
the existing 2-point diagonally implicit block backward differentiation
formula in terms of accuracy. Convergence is evident by the decrease
in error as the step length h tends to zero. Similarly, the solution at any
fixed point improves as the step length reduce. This can be seen from
the tables when h is reduced (from 0.01, 0.001., 0.0001, and 0.00001
to 0.000001). The maximum error indicates that the numerical result
becomes closer to the exact solution. Thus, the computed solution
tends to the exact solution as the step length tends to zero. Hence, the
new method converges faster for all the problems tested in comparison
with DI2BBDF.

h | Method | NS | MAXE | Time |
---|---|---|---|---|

10^{-2} |
DI2BBDF NDISBBDF |
1000 1000 |
6.85453e-002 7.15278e-002 |
2.47400e-001 1.57000e-001 |

10^{-3} |
DI2BBDF NDISBBDF |
1000 1000 |
2.60436e-002 2.32062e-003 |
1.58700e-001 1.609000e-001 |

10^{-4} |
DI2BBDF NDISBBDF |
100000 100000 |
2.84730e-003 2.68751e-005 |
3.23700e-001 9.01200e-001 |

10^{-5} |
DI2BBDF NDISBBDF |
1000000 1000000 |
2.87174e-004 2.74330e-007 |
1.14900e+000 5.11600e+000 |

10^{-6} |
DI2BBDF NDISBBDF |
10000000 10000000 |
2.87419e-005 2.75064e-009 |
9.85100e+000 5.44700e+001 |

**Table 1:** Numerical result for problem 1.

h | Method | NS | MAXE | Time |
---|---|---|---|---|

10^{-2} |
DI2BBDF NDISBBDF |
500 500 |
9.37034e+005 2.22919e-002 |
2.51600e-001 1.75400e-001 |

10^{-3} |
DI2BBDF NDISBBDF |
5000 5000 |
5.58180e-002 1.20098e-002 |
2.43700e-001 1.331000e-001 |

10^{-4} |
DI2BBDF NDISBBDF |
50000 50000 |
7.04562e-003 1.61824e-004 |
2.61300e-001 3.05800e-001 |

10^{-5} |
DI2BBDF NDISBBDF |
500000 500000 |
7.19659e-004 1.69025e-006 |
7.92400e-001 2.3500e+000 |

10^{-6} |
DI2BBDF NDISBBDF |
5000000 5000000 |
7.21171e-005 1.70019e-008 |
5.05000e+000 2.58100e+001 |

**Table 2:** Numerical result for problem 2.

h | Method | NS | MAXE | Time |
---|---|---|---|---|

10^{-2} |
DI2BBDF DIS2BBDF |
500 500 |
1.61797e+000 1.63063e-001 |
1.17000e-001 1.19600e-001 |

10^{-3} |
DI2BBDF DIS2BBDF |
5000 5000 |
1.45914e-001 3.27724e-003 |
1.09700e-001 1.50200e-001 |

10^{-4} |
DI2BBDF DIS2BBDF |
50000 50000 |
1.44486e-002 3.56827e-005 |
2.22200e-001 3.46700e-001 |

10^{-5} |
DI2BBDF DIS2BBDF |
500000 500000 |
1.44346e-003 3.60983e-007 |
7.39800e-001 2.89700e+000 |

10^{-6} |
DI2BBDF DIS2BBDF |
5000000 5000000 |
1.44332e-004 3.61514e-009 |
5.54800e+000 2.559000e+001 |

**Table 3:** Numerical result for problem 3.

The numerical results for the test problems given in section 6 are
tabulated in this section. The problems are solved using the new method
developed and the existing 2-point diagonally implicit block backward
differentiation formula developed by Zawawi [20]. The number of
steps taken to complete the integration and the maximum error for
the methods are presented and compared in **Tables 1-3**. In addition,
the graph of Log_{10} (MAXE) against h for each problem is plotted. The
notations used in the tables are listed below:

DI2BBDF=Diagonally Implicit 2-Point Block Backward Differentiation Formula

The graphs also show that the scaled errors for the new method are smaller when compared with that in the existing method.

A new method called New Diagonally Implicit Super Class of Block Backward Differentiation Formula (NDISBBDF) is developed. The order of the method is 2 and it is suitable for solving stiff IVPs. The stability analysis has shown that the method is both zero and A-stable. A comparison between the method and existing DI2BBDF is made and the results show that the method outperformed the existing DI2BBDF method in terms of accuracy.

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