A new 2-step fourth order implicit non-linear multistep method based on centroidal mean is considered in this paper. The new method is tested on some test problems; and numerical results show that the new method is able to produce acceptable numerical solutions for these test problems. Comparisons in terms of numerical accuracy between the new method and the classical 2-step Adams-Moulton method are carried out as well. Numerical experiments show that our new method performs better than the classical 2-step Adams-Moulton method in solving these test problem

Teh, Yuan Ying

Yaacob, Nazeeruddin

English

A new 2-step fourth order implicit non-linear multistep method based on centroidal mean is considered in this paper.The new method is tested on some test problems; and numerical results show that the new method is able to produce acceptable numerical solutions for these test problems.Comparisons in terms of numerical accuracy between the new method and the classical 2-step Adams-Moulton method are carried out as well. Numerical experiments show that our new method performs better than the classical 2-step Adams-Moulton method in solving these test problems

UUM Repository

A new non-linear multistep method based on centroidal mean

in solving initial value problems

Universiti Teknologi Malaysia Institutional Repository

MATEMATIKA, 2009, Volume 25, Number 2, 167–176c©Department of Mathematics, UTM.A New Non-linear Multistep Method Based on Centroidal Meanin Solving Initial Value Problems1Nazeeruddin Yaacob & Teh Yuan YingDepartment of Mathematics, Faculty of Science, Universiti Teknologi Malaysia81310 UTM Skudai, Johor, Malaysiae-mail: 1ny@mel.fs.utm.myAbstract A new 2-step fourth order implicit non-linear multistep method based oncentroidal mean is considered in this paper. The new method is tested on some testproblems; and numerical results show that the new method is able to produce accept-able numerical solutions for these test problems. Comparisons in terms of numericalaccuracy between the new method and the classical 2-step Adams-Moulton methodare carried out as well. Numerical experiments show that our new method performsbetter than the classical 2-step Adams-Moulton method in solving these test problems.Keywords Initial value problems, non-linear multistep method; centroidal mean;Adams-Moulton method; 3-stage fourth order Lobatto IIIC method.1 IntroductionNumerical methods from the class of linear multistep methods and the class of Runge-Kuttamethods are defined by [1]k∑j=0αjyn+j = hk∑j=0βjfn+j (1)andyn+1 = yn + hs∑i=1bikiki = f(tn + cih, yn + hs∑j=1aijkj), i = 1, 2, . . . , s.(2)respectively. These methods are among the most common used numerical methods for thefirst order initial value problem of the formy′ (t) = f (t, y (t)) , y (a) = y0, t ∈ [a, b] . (3)A new research trend had emerged around the 1990’s where researchers start to incorporatemean expressions into linear Runge-Kutta method in (2) to form a new kind of Runge-Kutta method that based on different kinds of means. This special type of Runge-Kuttamethod is considered as non-linear method due to the non-linear structures that arise fromthe implementation of various mean expressions. Articles which have discussed this type ofmethod are such as [3] through [28].In this article, we shall explore the possibility of deriving multistep method based onmean expressions for the numerical solution of (1). Our study is motivated by the successof [14] and [22] in deriving several non-linear multistep methods based on different meanexpressions for the numerical solution of second order initial value problem. This article is168 Nazeeruddin Yaacob & Teh Yuan Yingorganized as follows: In section 2, we present the procedure for obtaining the new 2-stepimplicit non-linear method based on centroidal mean. In Section 3, we present the localtruncation error, consistency, zero-stability and convergence analysis of the new method.The stability polynomial and the regions of absolute stability for the new method arepresented in Section 4. Section 5 shows the numerical implementations of the new implicitmethod to a variety of test problems and compares its performance with the classical 2-step implicit Adams-Moulton method in terms of numerical accuracy. Some remarks andconclusions will be given in Section 6.2 Derivation of the 2-step Implicit Non-linear Method Based onCentroidal MeanFirstly, we define the new 2-step implicit method asα2yn+2 + α1yn+1 + α0yn = h(c1fn+2 + c2fn+1 + c3fn + c42(f2n+2+fn+2fn+1+f2n+1)3(fn+2+fn+1)+c52(f2n+1+fn+1fn+f2n)3(fn+1+fn)+ c42(f2n+2+fn+2fn+f2n)3(fn+2+fn)) (4)where α0 = 0, α1 = −1, α2 = 1 with c1, c2, c3, c4, c5 and c6 are constants that need to bedetermined. Note that fn+2+fn+1 6= 0, fn+1+fn 6= 0 and fn+2+fn 6= 0. On using Taylorseries to expand both sides of equation (4) up to O(h4), and compare each coefficient, weobtain the following equations:c1 + c2 + c3 + c4 + c5 + c6 = 1, (5)2c1 + c2 +32c4 +12c5 + c6 =32, (6)2c1 +12c2 +54c4 +14c5 + c6 =76, (7)112c4 +112c5 +13c6 = 0, (8)−18c4 −124c5 −13c6 = 0, (9)14c4 +112c5 +23c6 = 0. (10)Using MATHEMATICA 5.0 in solving the system of equations given in (5) – (10), we obtaina set of solutions in terms of a free parameter c6 shown as follows:c1 =512+c62, c2 =23+ 2c6, c3 = −112+c62, c4 = −2c6, c5 = −2c6.On substituting these ci, i = 1, 2, . . . , 6 and α0 = 0, α1 = −1, α2 = 1 into equation (4), theresulting method is a 2-step implicit non-linear method based on centroidal mean:yn+2 − yn+1 = h((512+ c62)fn+2 +(23+ 2c6)fn+1 +(− 112+ c62)fn+ (−2c6)2(f2n+2+fn+2fn+1+f2n+1)3(fn+2+fn+1)+ (−2c6)2(f2n+1+fn+1fn+f2n)3(fn+1+fn)+c62(f2n+2+fn+2fn+f2n)3(fn+2+fn)) (11)A New Non-linear Multistep Method Based on Centroidal Mean 169Method (11) is named Non-linear Multistep method based on centroidal mean of two stepsand fourth order or shortly NLMMCeM(2,4). The local truncation error in terms of c6 forNLMMCeM(2,4) is given byh4(−124f ′′′n)+ h5(c6 (f′n)412 (fn)3 −c6 (f′n)2f ′′n12 (fn)2 +c6 (f′′n )212fn−17360f(4)n)+O(h6). (12)Since there is a free parameter c6, we choose this parameter so that the local truncationerror shown in (12) is in O(h6). From (12), we force the first two terms to zero that ish4(−124f ′′′n)+ h5(c6 (f′n)412 (fn)3−c6 (f′n)2f ′′n12 (fn)2+c6 (f′′n )212fn−17360f(4)n)= 0. (13)After some algebraic manipulations, c6 is obtained as follows:c6 =(fn)3(15f ′′′n + 17hf(4)n)30h((f ′n)4− fn (f ′n)2f ′′n + (fn)2(f ′′n )2) (14)where (f ′n)4− fn (f′n)2f ′′n + (fn)2(f ′′n )26= 0. Note that c6 is a constant since all functionsf(i)n+j , i = 0, 1, 2, 3, 4, 5are evaluated at the point tn. Therefore, the local truncation errorfor NLMMCeM(2,4) with c6 in (14) is given byLTE (Centroidal)=h5720−2(3f ′n((f ′n)2− fnf′′n)2+ (fn)2((f ′n)2− 2fnf′′n)f ′′′n)(15f ′′′n + 17hf(4)n)fn((f ′n)4− fn (f ′n)2f ′′n + (fn)2(f ′′n )2)− h6(7240f(5)n)+O(h7). (15)3 Consistency, Stability and Convergence Analysis for NLMM-CeM(2,4)We extend the theory of consistency, zero-stability and convergence for the linear multistepmethod to the new method NLMMCeM(2,4). As usual, the first characteristic polynomial,ρ (ζ) and the second characteristic polynomial of NLMMCeM(2,4), σ (ζ) can be obtainedfrom the left-hand side and right-hand side of equation (11) respectively; with the substi-tution of yn+j = fn+j = ζj and f(i)n+j = ζj for i = 0, 1, 2, 3, 4, 5 and j = 0, 1, 2. Therefore,we obtainρ (ζ) = ζ2 − ζ (16)andσ (ζ) =(512 +c62)ζ2 +(23 + 2c6)ζ +(− 112 +c62)+ (−2c6)2“(ζ2)2+ζ2×ζ+ζ2”3(ζ2+ζ)+(−2c6)2(ζ2+ζ×1+12)3(ζ+1)+ c62“(ζ2)2+ζ2×1+12”3(ζ2+1)(17)170 Nazeeruddin Yaacob & Teh Yuan YingFrom the assumption f(i)n+j = ζj for i = 0, 1, 2, 3, 4, 5, we note that c6 in (14) is evaluatedat the point tn and therefore we have f(i)n = ζ0 = 1 for i = 0, 1, 2, 3, 4, 5. On substitutingf(i)n = ζ0 = 1 for i = 0, 1, 2, 3, 4, 5 into equation (14) yieldc6 =(13)(15 (1) + 17h (1))30h((1)4− 1× (1)2× 1 + (1)2(1)2) = 15 + 17h30h.The first derivative of equation (16) isρ′ (ζ) = 2ζ − 1. (18)On substituting ζ = 1 into equations (16), (17) and (18) we obtain the following results:ρ (1) = 0, σ (1) = 1 and ρ′ (1) = 1. (19)Since conditions in (19) hold for NLMMCeM(2,4), then we can say that it is consistent.To determine the zero-stability of NLMMCeM(2,4), we must make sure that no root ofρ (ζ) has modulus greater than one, and every root with modulus one is simple. Therefore,from (16), the roots ofζ2 − ζ = 0are ζ1 = 1 and ζ2 = 0. Consequently, we have |ζ1| = 1 and |ζ2| = 0 which are not greaterthan one and simple. In view of this, we can say that NLMMCeM(2,4) is zero-stable.Finally, we can claim that NLMMCeM(2,4) is convergent because it is shown to beconsistent and zero-stable.4 Absolute Stability of NLMMCeM(2,4)In order to carry out the stability analysis for NLMMCeM(2,4), we must obtain the stabilitypolynomial and its corresponding regions of absolute stability. We can obtain the stabil-ity polynomial of NLMMCeM(2,4) by applying the Dahlquist’s test equation y′ = λy toequations (11) and (14) [2]. Note that λ is a complex constant with negative real part. Onsubstituting (14) into (11) and then substituting fn+2 = λyn+2, fn+1 = λyn+1, fn = λyn,f ′n = λ2yn, f′′n = λ3yn, f′′′n = λ4yn, f(4)n = λ5yn, yn+2 = ζ2, yn+1 = ζ and yn = 1 into (11),we obtain the following stability polynomial for NLMMCeM(2,4) as follows:(195− 58z) ζ4 − (240 + 188z) ζ3 + (270 + 42z) ζ2 − (240 + 188z) ζ + (15 + 32z) = 0 (20)where z = hλ. Here, ζ can be interpreted as the characteristic roots of the differenceequation (11). The condition for the stability is that the roots of (20) i.e. ζ are all ofabsolute value less than 1. By taking z as complex number i.e. z = x + iy, we plot theregion which satisfies the condition that all roots of (20) are of absolute value less than 1in Figure 1. The shaded region in Figure 1 is the region which satisfies the condition thatall roots of (20) are of absolute value less than 1. Consequently, the shaded region is theregion of absolute stability of NLMMCeM(2,4).A New Non-linear Multistep Method Based on Centroidal Mean 171Figure 1: Stability Region of NLMMCeM(2,4)5 Numerical Experiments and ComparisonsIn this section, NLMMCeM(2,4) is used to solve some test problems in order to check itsreliability and accuracy. We present i) the maximum absolute error over the integrationinterval given bymax0≤n≤N{|y (tn)− yn|}where N is the number of integration steps; and ii) the absolute error at the end-point ofintegration interval given by |y (tn)− yN | for each test problem. Note that y (tn) representsthe exact solution of a test problem at point tn, while yn is the approximations of the exactsolution at point tn of a test problem. The notation 1.26681(-5) indicates 1.26681× 10−5.Numerical results obtained using NLMMCeM(2,4) is compared with the numerical resultsobtained using the classical 2-step implicit Adams-Moulton method given by [29]yn+2 − yn+1 = h(512fn+2 +23fn+1 −112fn)(21)where the local truncation error of (21) isLTE (Adams−Moulton) = h4(−124f ′′′n)+ h5(−17360f(4)n)+O(h6). (22)The starting values y1 for NLMMCeM(2,4) and 2-step Adams-Moulton are computed viathe 3-stage fourth order Lobatto IIIC method shown in the following Butcher tableau [30]:172 Nazeeruddin Yaacob & Teh Yuan Ying016−13161216512−1121162316162316Problem 1: [17]y′ (t) = y (t)− t2 + 1, y (0) =12, t ∈ [0, 1] .The exact solution for Problem 1 is given by y (t) = (t+ 1)2− 12et.Problem 2: [24]y′ (t) = y (t) cos t, y (0) = 1, t ∈ [0, 1] .The exact solution for Problem 2 is given by y (t) = esin(t).Problem 3: [31]y′1 (t) = 0.2y2 (t) , y1 (0) = 1,y′2 (t) = −0.2y1 (t) , y2 (0) = 1.Problem 3 is solved numerically over the integration interval t ∈ [0, 1] and the exact solutionsfor Problem 3 are given by y1 (t) = cos 0.2t+ sin 0.2t and y2 (t) = − sin 0.2t+ cos 0.2t.Table 1 to Table 8 show that NLMMCeM(2,4) has no difficulty in solving all the testproblems mentioned above; and it performs better than 2-step Adams-Moulton for differentstep length for Problem 1 and Problem 2. On the other hand, NLMMCeM(2,4) givescomparable accuracy to 2-step Adams-Moulton method in solving Problem 3 which is asystem of first order differential equations.Table 1: Maximum Absolute Error for Problem 1With Respect to Step Length, hh Adams-Moulton NLMMCeM(2,4)1/16 1.26681(−05) 8.64014(−07)1/32 1.65514(−06) 5.55726(−08)1/64 2.11439(−07) 5.17533(−09)1/128 2.67161(−08) 4.32471(−10)A New Non-linear Multistep Method Based on Centroidal Mean 173Table 2: Error at the End-point for Problem 1With Respect to Step Length, hh Adams-Moulton NLMMCeM(2,4)1/16 1.26681(−05) 8.64014(−07)1/32 1.65514(−06) 5.55726(−08)1/64 2.11439(−07) 5.17533(−09)1/128 2.67161(−08) 4.32471(−10)Table 3: Maximum Absolute Error for Problem 2With Respect to Step Length, hh Adams-Moulton NLMMCeM(2,4)1/16 6.12206(-05) 1.36354(-05)1/32 7.90321(-06) 9.23572(-07)1/64 9.98723(-07) 9.06451(-08)1/128 1.24430(-07) 9.36788(-09)Table 4: Error at the End-point for Problem 2With Respect to Step Length, hh Adams-Moulton NLMMCeM(2,4)1/16 6.12206(−05) 1.36354(−05)1/32 7.90321(−06) 9.23572(−07)1/64 9.98723(−07) 9.06451(−08)1/128 1.24430(−07) 9.36788(−09)Table 5: Maximum Absolute Error for Problem 3With Respect to Step Length, h(y1(t))h Adams-Moulton NLMMCeM(2,4)1/16 4.63483(−06) 4.63637(−06)1/32 1.15738(−06) 1.15747(−06)1/64 2.89350(−07) 2.89355(−07)1/128 7.23379(−08) 7.23382(−08)Table 6: Error at the End-point for Problem 3With Respect to Step Length, h(y1(t))h Adams-Moulton NLMMCeM(2,4)1/16 4.52400(−06) 4.59524(−06)1/32 1.13190(−06) 1.14109(−06)1/64 2.83275(−07) 2.8444(−07)1/128 7.08570(−08) 7.10037(−08)174 Nazeeruddin Yaacob & Teh Yuan YingTable 7: Maximum Absolute Error for Problem 3With Respect to Step Length, h(y2(t))h Adams-Moulton NLMMCeM(2,4)1/16 9.37627(−07) 9.08091(−07)1/32 2.32079(−07) 2.28239(−07)1/64 5.77552(−08) 5.72659(−08)1/128 1.44052(−08) 1.43435(−08)Table 8: Error at the End-point for Problem 3With Respect to Step Length, h(y2(t))h Adams-Moulton NLMMCeM(2,4)1/16 9.37627(−07) 9.08091(−07)1/32 2.32079(−07) 2.28239(−07)1/64 5.77552(−08) 5.72659(−08)1/128 1.44052(−08) 1.43435(−08)6 ConclusionsWe have presented a new 2-step fourth order non-linear multistep method based on cen-troidal mean (NLMMCeM(2,4)), that is suitable to solve first order initial value problems.Classical 2-step Adams-Moulton method is a third order method, but NLMMCeM(2,4) canachieved fourth order of accuracy by choosing the appropriate parameter c6. This newmethod is shown to be consistent, zero-stable and convergent. Numerical results presentedin Section 5 also suggest that NLMMCeM(2,4) is suitable to solve both single differentialequation and systems of first order differential equations.AcknowledgementThe authors would like to thank Universiti Teknologi Malaysia (UTM) for providing fundunder the grant vot 78337 to conduct this research.References[1] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley &Sons, Chichester, 1991.[2] J.D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley& Sons, London, 1973.[3] D.J. Evans & B. B. Sanugi, A comparison of numerical ODE solvers based on arithmeticand geometric means, Int. J. Comput. Math. 23(1987), 37–62.[4] D. J. Evans & B. B. Sanugi, A nonlinear Runge-Kutta formula for initial value prob-lems, SIGNUN Newsletter 22 (3)(1987), 27 – 30.A New Non-linear Multistep Method Based on Centroidal Mean 175[5] B.B. 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A new non-linear multistep method based on centroidal mean in solving initial value problems

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A new non-linear multistep method based on centroidal mean in solving initial value problems

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