A Modified Block Adam Moulton ( MOBAM ) Method for the Solution of Stiff Initial Value Problems of Ordinary Differential Equations

Stiff ordinary differential equations pose computational difficulties as they present severe step size restrictions on the numerical methods to be used. Construction of numerical methods that possess suitable stability properties for the solution of such systems has been the target of many researchers. Development of methods suitable for these systems of equations has been either through the use of derivative of the solution or by introducing off-step points, additional stages or super future points. These processes have been exploited in Runge-Kutta methods or linear multistep methods. In this study, an improved class of linear multistep block method has been constructed based on Adams Moulton block methods. The improved methods are shown to be A-stable, a property desirable to handle stiff ODEs. Methods of uniform orders 10 and 11 have been constructed. The efficiency of the new methods tested on stiff systems of ODEs and the results reveal that the MOBAM methods compare favourably with results obtained using the state of the art Matlab Ode23 solver.


INTRODUCTION
Stiff initial value problems of ordinary differential equations were realized in 1952 arising from the study of chemical kinetics.Since then, many researches have been involved in the study and development of suitable numerical methods to handle them.Curtiss and Hirschfelder (1952), Mitchell and Craggs (1953) and Gear (1971) developed the Backward Differential Formulae which were used to solve stiff problems arising from chemical kinetics.Intensive work done in this regard yielding several efficient numerical algorithms has appeared in the literature.Because of the severe restriction on step size placed on Adams Moulton methods for the solution of stiff ODEs, Garfinket et al. (1978) introduced the concept of Astability for linear multistep methods.This property became the minimum requirement for any linear multistep method to be used for the solution of stiff ODEs.Achieving A-stability was difficult, other stability requirements such as A(α)-stability and stiffly stability were considered.Chakrararti and Kamel (1983) developed stiffly stable second derivative methods with high order and improved stability regions to handle stiff problems.Several other researchers such as Samir (2013), Evelyn and Renante (2006), Song (2010), Vlachos et al. (2009) and Ali and Gholamreza (2012) developed improved and more friendly numerical methods with improved stability properties that could handle stiff ODEs with various degrees of stiffness.
In this study, we pursue the path of Adams Moulton methods by constructing A-stable block linear multistep methods of uniform order 10 and 11.These methods are obtained by modifying the Adams Moulton method of step number 9 by reducing its interpolation step number from 8 to 3.This approach is similar to that of Brugnano and Trigiante (1998) where they developed the Generalized Adams methods.
The modified methods are constructed using the concept of multistep collocation of Lie and Norsett (1989) which Onumanyi et al. (1994Onumanyi et al. ( , 1999) ) referred to as block linear multistep methods.The block methods are arrived at by evaluating the continuous formulation of the new method at grid and off grid points to yield the discrete schemes used as block integrators.The methods are applied simultaneously producing selfstarting methods thus eliminating the issue of overlap of pieces of solutions.

CONSTRUCTION OF THE NEW METHOD
The k-step general linear multistep method for numerically solving the differential equation: being the continuous coefficients of the method.
The new improved nine step method based on the Adams Moulton method is defined as: where, Using the method in Onumanyi et al. (1994Onumanyi et al. ( , 1999) ) and further studied by Kumleng et al. (2012) and Chollom et al. (2012) the method (3) for 2 1 − = k v and j =0, 1… 9 sequence in the matrix form: where, C being the elements of the continuous coefficients of the method.

Construction of MOBAM K = 9:
This method has its general form as: Using the procedure in Onumanyi et al. (1999), we obtain the D matrix for (5) as: Using the Maple software, the inverse of the matrix ( 6) is obtained and represented as C, the elements of which yield the continuous coefficients α 3 (x) and β j (x), j = 0,1,…9 of the method (5) as: Substituting the continuous coefficients (7) into (5) yields the continuous form of the MOBAM method (8): . Evaluating the continuous scheme (8) at the following points yields the MOBAM method ( 9) for k = 9 used in block for the solution of ODEs: : This is the hybrid form of the MOBAM family for k=9.An off step point is inserted in (5) to give its general form as: Following a similar procedure as in above section gives the matrix: The continuous coefficients of the method (10) are similarly obtain from the inverse of (11) as:

) ( h h h h h h h h h h h x h h h h h h h h h h h h x h h h h h h h h h h h h x h h h h h h h h h h h h x h h h h h h h h h h h h x h h h h h h h h h h x h h h h h h
Substituting the continuous coefficients ( 12) into (10) produces the continuous form of the new MOBAM method for k = 9, 2 17 = µ as: . Evaluating the continuous scheme (13) at the following points µ yields the discrete members of the Modified Block Hybrid Adams Method (MOBHAM) used as block integrators as:

STABILITY ANALYSIS OF THE NEW BLOCK METHODS
The block method ( 9) is represented by a matrix finite difference equation in block form as:  being 9 by 9 matrices whose entries are given by the coefficients of ( 9) and are as defined in equation ( 16) below: where, 0,1,2,...9 w = and n is the grid index.
Zero-stability: Zero-stability is concerned with the stability of the difference system (15) as h tends to zero.Thus, as 0 h → , the method (15) tends to the difference system: By Fatunla (1991), the block method ( 9) is zero-stable since in (19), , the multiplicity does not exceed 1.The block method is therefore convergent according to Henrici (1962).Since the MOBAM method is both consistent and zero-stable, it is convergent.Using the same procedure, the MOBHAM method in ( 14) is also zero-stable and consistent, hence convergent.

Order of the MOBAM methods:
The orders and error constants of the new block methods are obtained using Chollom et al. (2007).The block method (10) is of uniform order   Butcher (1985), the block methods ( 9) and ( 14) are reformulated as General linear methods and the region of absolute stability of the method was plotted using the matlab program and is shown in Fig. 1.
Figure 1 reveals that the MOBAM methods are Astable.

NUMERICAL EXPERIMENTS
In this section, the MOBAM methods ( 9) and ( 14) are tested on linear and nonlinear stiff initial value problems of ODEs.The solution curves obtained for the MOBAM methods are compared with the well-known MATLAB ODE 23 solver.Example 1: We consider a well -known classical system experimented in Baker (1989), Stabrowski (1997) Its exact solution is given by the sum of two decaying exponential components: The stiffness ratio is 1:1000, h = 0.1 (Fig. 2 and 3) Example 2: Consider the stiffly nonlinear problem which was proposed by Kaps et al. (1981)  The smaller is the ε , the more serious the stiffness of the system.Its exact solution is given by (Fig. 4 and 5): The stiffness ratio of the problem is 1000.h = 0.1, 0≤x≤10.Its exact solution is given by (Fig. 6 and 7):