369 Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 2017 pp

Size: px

Start display at page:

Download "369 Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 2017 pp"

Coral Hunter
6 years ago
Views:

1 369 Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 27 pp Original Research Article THIRD DERIVATIVE MULTISTEP METHODS WITH OPTIMIZED REGIONS OF ABSOLUTE STABILITY FOR STIFF IVPS IN ODES * Athe, B.O. and Mua, K.O. Advanced Research Laboratory, Department of Mathematics, Faculty of Physical Sciences, University of Benin, PMB 54, Benin City, Nigeria. *blessingathe@gmail.com ARTICLE INFORMATION Article history: Received 3 October, 27 Revised November, 27 Accepted 4 November, 27 Available online 29 December, 27 Keywords: Characteristics polynomials Region of absolute stability Adam s type A- stable A(α) stable. INTRODUCTION ABSTRACT Adam s type methods are nown to be zero-stable by design. The bacward differentiation formulas are viewed as the dual of the Adams method because of the structure of their second and third characteristics polynomials. Although they are plagued by zero-instability for large step sizes, they are good integrator for stiff initial value problems in ordinary differential equations. This paper is on the derivation of method which combines the characteristics of Adam s type methods and the bacward differentiation formulas using the methods of collocation and interpolation. Proposed method is A-stable for order p 7 and A(α)-stable for p 2. Numerical eamples are presented to show the suitability of method developed in the integration of stiff initial value problems. 27 RJEES. All rights reserved. Consider the initial value problems (IVPs) in ordinary differential equation (ODE) of the form: ( ) ( ( )) ( ) y = f, y, y = y, () m on the finite interval I = [ ] where y : [, N ] R and [ N ], N f :, XR m R m where f is continuous and twice differentiable, which often aid the formation of Mathematical models of real life phenomena.solution to Equation () is often times insoluble using analytical means, hence the need for numerical techniques in solving such problems. The numerical methods for solving IVPs in ODEs lie Equation () are classified into one-step (multistage) methods and multistep (one stage) methods (Hairer and Wanner, 22), The Runge-Kutta

2 B.O. Athe and K.O. Mua / Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 27 pp methods belong to the former group, while Adams-Bashforth and Adams-Moulton methods are members of the later (Hairer and Wanner, 22). Multiderivative methods for solving systems of ODEs proposed in Enright (974) and Cash (98) incorporate higher derivative into their formula. The ustification for including higher derivative term into methods is in Enright (974) and this includes to obtain stability at infinity with reasonable wide stability region in neighborhood of the origin. In Enright (974), a second derivative linear multistep method was constructed, and this method is of order p = +2 for a -step. In this paper, a class of third derivative implicit Adam s type Multistep methods for solving stiff IVPs in ODEs is developed. 2. THIRD DERIVATIVE LINEAR MULTISTEP METHODS (TDLMMs) The general Third Derivative Linear Multistep Methods (TDLMMs) is of the form: α y = h β f h γ g h δ l n+ + 2 n+ + 3 n+ n+ = = = = (2) where α, β, γ, δ are real parameters to be determined. If any or all of the parameters β, γ and δ are non-zero, the TDLMMs in (Equation 2) is said to be implicit, else it is eplicit. The first, second, third and fourth characteristics polynomials associated with TDLMM in (Equation 2) are given as: ρ ζ = α ζ ; σ ζ = β ζ ; η ζ = γ ζ ; π ζ = δ ζ ( ) ( ) ( ) ( ) = = = =. (3) The order conditions for TDLMM in (Equation 2) is: q q q q C = α q β q(q ) 2 γ q(q )(q ) 3 γ, q =,,2,..., p q q! 2 (4) = The TDLMM in (Equation 2) is of order p, if C = q for q p and C p + is the principal error constant. In Ezzeddinne and Hoati, (22), a family of Third derivative multistep methods (TDMM) of the form in Equation 5 was considered and the method was stable for order p and unstable otherwise. It was designed to bypass stability constraints imposed by the Dahlquist s order barrier theorem (Dahlquist, 963). 2 3 = α + β + γ n+ n+ n+ n+ n+ = y y h f h g h l The method in Equation (5) has same first, second and third characteristics polynomial with those developed in (Hairer and Wanner, 22). In situation where methods with high order of accuracy are required, higher derivative methods have been proven to be search direction for the development of high order numerical method for integrating IVPs in (Equation ). The third derivative multistep methods in Ezzeddinne and Hoati (22) are however inefficient in situations for which methods with order as high as p =2 is required. To derive a more efficient method compared to Equation (5) in terms of order of accuracy, a non-zero coefficient will be inputted into the third characteristics polynomial of TDMM in Equation (5). This idea was utilized in Mua and Obiorah (26) to improve the efficiency of the second derivative bacward differentiation formula (SDBDF). The term is (5)

3 B.O. Athe and K.O. Mua / Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 27 pp introduced in such a way that -coefficients previously set to zero in the construction of TDMM in Equation (5) is assumed non-zero, and this results in the development of a linear multistep method for each -step method. Herein, a third derivative linear multistep method of the form in Equation 6 is proposed. ( µ µ ) 2 3 n+ n+ = β n+ + γ n+ + γ n+ + δ n+ = y y h f h g g h l The term γ -µg n+- added to Equation (5) leads to the increasing order of method in Equation (6) by one compared with that of TDMM in Equation (5), where the term γ -µ, µ=,2,, for each µ value is assumed non-zero. The parameters δ, γ, γ µ, µ=() and β, = ( ) are real constants to be determined. 3. CONSTRUCTION OF THE METHOD The collocation and the interpolation methods were used in the construction of the new third derivative multistep method in Equation (6) as follows: Given the power series polynomial, a basic function to approimate the solution of the IVPs in Equation (). ( ) y N = = a where a are the unnown coefficients and are the polynomial function, N=+4 is the degree of the polynomial. Differentiating Equation (7) with respect to results in: ( ) = + = 4 = y f a = 2 ( ) = ' = = ( ) y '' f g a (9) = 3 ( ) = = = = ( )( ) y ''' f '' g ' l 2 a. () Interpolating Equation (7) at, n n + = = and collocating Equation (8), Equation (9) and Equation () at =, =, 2,..., n+, = n +, = n+ µ and = n + respectively results in the system of linear equations: (6) (7) (8) L n+ n+ n+ n+ n+ a y n L ( + 4) a f n n n n n L ( + 4) a f n+ n+ n+ n+ 2 n+ M M M M M M M M = M L ( + 4) a f n+ n+ n+ n+ + n L ( + 4)( + 3) a g n+ n+ n+ + 2 n L ( + 4)( + 3) a g n+ µ n+ µ n+ µ + 3 n+ µ + ( + )( + )( + ) a l 6 24 L n+ n+ + 3 n+ ()

4 B.O. Athe and K.O. Mua / Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 27 pp To determine the values of the a s, the system in Equation () was solved using the Gaussian elimination method, and afterwards the a s was substituted into Equation (7) with = n+t, α = to yield a continuous linear multistep formulas. The resulting scheme was then evaluated at given values of t to obtain the discrete coefficients for fied. For each, we select the µwith the largest α-value and in circumstances where the α-values are the same the one with the smallest error constant is selected as the optimal region of absolute stability. The coefficients of the proposed method in Equation (6) are presented in appendi. 4. STABILITY ANALYSIS The stability behavior of the new method is eamine by applying Equation (6) to the scalar test equation in Equation (2) to give Equation (3) a polynomial of degree three in z. y = λ y, λ C µ ζ = ( zβ z γ z δ ) ( zβ ) ζ z β ζ z γ ζ µ (3) = 2 2 i =. If we set ξ = e π θ, θ < where z λh in Equation (3), using the boundary locus method, the roots of the polynomial in Equation (3) describe the stability region R A of the proposed method in Equation (6). In eamining the stability properties of the proposed method in Equation (6), the boundary locus method was used to obtain the boundary describe for each step. The proposed method in Equation (6) was found to be A-stable for 3 for all values of µ and A(α)-stable for =3,4,, 8 for all µ and unstable for >8. The angle of absolute stability including the order and error constants of the proposed TDLMM in Equation (6) and that of TDLMM constructed in (Ezzeddinne and Hoati, 22), Equation (5) are presented in Tables and 2 respectively. In Table, the stability characteristics of method proposed in this paper are presented, while in Table 2 is the characteristics of the TDMMs of (Ezzeddinne and Hoati, 22). The proposed methods in Equation (6) are of higher order with smaller error constants when compared with TDMMs in Equation (5). Both methods are A-stable for stepsizes =,2,3. Also, proposed method in Equation (6) has a wider stability region R A compared with TDMMs in Equation (5). Table : Optimized stability characteristics, order and error constants of Equation (6) μ p α 9 o 9 o 9 o 89.9 o 88 o 8 o 78 o 75.5 o C p (2) Table 2: Stability characteristics, order and error constants of TDMMs in Equation (5) μ p α 9 o 9 o 9 o 89.9 o 89. o 77 o 77 o 65 o C p

5 B.O. Athe and K.O. Mua / Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 27 pp NUMERICAL EXPERIMENTS In this section, the first member of the proposed TDMM in Equation (6) is applied on three standard stiff initial value problems to illustrate the suitability of the proposed methods in the integration of stiff and non-stiff IVPs. The results obtain were compared with the analytic solution of the problems. The starting value for the method was obtained using the second derivative eplicit one-step method., 2 y = y + hf + h f (4) The following standard problems are considered for the numerical eperiments: Problem. Source: Kaps (98) Consider the non-linear system of IVPs y = ε + 2 y + ε y y = ( ) ; ( ) 2 2 y = y y ; y () =, n+ n n 2 n 2 Forε=2. Eact solution is given by: t t 2 Table 3: Numerical solution of problem () generated by third derivative method in Equation (6) t y i TDLMM (6) Eact Error Y()-y n.4.4 h =. y = e, y = e y e e e-4 y e e e-6 y e e e-3 y e e e-3 Problem 2: Source : Detest class A 2 Consider the nonlinear problem y y =, y() =, t [,4], h = Eact solution y = t + Table 4: Numerical solution, problem (2) generated by proposed method in Equation (6) t TDLMM (6) y n Eact Y(t n) Error Y(t n)-y n e e e e e e e-.966e e-3 Problem 3: (CF: Ouonghae and Nwoorie, 24) Consider the linear system of stiff IVP y = 8y + 7y, y() = 2 ( ) y = 42y 43y, y = [, ], ( ) 5, Eact solution is y = 2e e, h =. 5 y2 ( ) = 2e + 6e

6 B.O. Athe and K.O. Mua / Nigerian Research Journal of Engineering and Environmental Sciences 2(2) 27 pp CONCLUSION Table 5: Numerical solution, problem (3) generated by proposed method in Equation (6) t y i TDLMM (6) y n Eact Y(t n) Error Y(t n)-y n.4 y e-4 y e-4.4 y e-5 y e-5 4 y e e e-6 y e e e-6 A new class TDLMM is derived by the addition of non-zero term in the third characteristics polynomial of the third derivative multistep methods (TDMMs) derived in (Ezziddenne and Hoati, 22). The boundary locus method is used to select stable TDLMM with the largest region of absolute stability. Methods proposed herein is of order one higher than that of TDMMs developed in (Ezziddenne and Hoati, 22). This class of method combines the ecellent properties of Adam s type method and TDBDF of being zero-stable by design and stable at infinity respectively. TDMMs in Equation (5) is unstable for order p, but methods proposed in this paper is stable for order upto p CONFLICT OF INTEREST There is no conflict of interest associated with this wor. SUPPLEMENTARY INFORMATION Table S: Coefficients of -step of the proposed TDMM in Equation (6) for each µ. This material is available free of charge via the Internet at REFERENCES Cash, J.R. (98). Second derivative etended bacward differentiation formula for the numerical integration of stiff system. Society for Industrial and Applied Mathematics Journal. Numerical Analysis,8(5), pp Dahlquist, G. (963). A special stability problem for linear multistep methods. BIT 3, Enright, W.H. (974). Second derivative multistep methods for stiff ordinary differential equations. Society for Industrial and Applied Mathematics Journal. Numerical Analysis,, pp Ezzeddinne, A.K. and Hoati, G. (22). Third derivative multistep methods for stiff systems. International Journal of Nonlinear science, 4, pp Hairer, E. and Wanner, G. (22). Solving ordinary differential equation II. Stiff and differential Algebraic problems. Vol. 2, Springer-verlag, Berlin Heidelberg. Kaps, P. (98). Rosenbro-types methods in: Numerical methods for stiff initial value problems (eds: Dalquist, G and Jeltsch, R). Bericht Nr. 9, institute fur geometric and pratische Mathemati der RWTH Aachen. Mua, K.O. and Obiorah, F.O. (26). Boundary locus search for stable second derivative linear multistep methods for stiff initial value problems in ordinary differential equation. Journal of the Nigeria Association of Mathematical Physics, 37, pp Ouonghae R.I. and Ihile M.N.O. (22). A continuous formulation of A(α) -stable second derivative linear multistep methods for stiff initial value problems in ordinary differential equations. Journal of Algorithm And Computational Technology, 6(), pp.79-. Ouonghae R.I. and Nwoorie, N.J. (24). A modified third derivative linear multistep method for stiff ODEs. Journal of the Nigeria Association of Mathematical Physics, 28, pp. 7-4.

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

A Family of L(α) stable Block Methods for Stiff Ordinary Differential Equations American Journal of Computational and Applied Mathematics 214, 4(1): 24-31 DOI: 1.5923/.acam.21441.4 A Family of L(α) stable Bloc Methods for Stiff Ordinary Differential Equations Aie I. J. 1,*, Ihile