Obrechkoff Versus Super - Impl ic i t M e t h o d s for the Solut ion of First - and Second - Order Init ial Value

K e y w o r d s O b r e c h k o f f methods, Super-implicit method, Initial value problems. 1: I N T R O D U C T I O N In th is paper , we discuss t h e numer ica l so lu t ion of f i rs t -order ini t ia l va lue p rob l ems ( IVPs) y'(x) = f ( x , y(~)) , y(0) = y0, (1) and a specia l class (for which yl is miss ing) of second-order I V P s y ' ( x ) = f ( z , y ( x ) ) , y(O) = Yo, yI(O) = Y~. (2) T h e r e is a vas t l i t e r a tu re for t h e numer ica l so lu t ion of these p rob l ems as well as for t h e genera l second-order I V P s y"(x) = : ( x , y (z ) , ~'(~)) , y(0) = y0, ~'(0) = y~. (3) See, for example , t h e excel lent book by L a m b e r t [1]. Here, we are in te res ted specif ical ly in two classes of me thods . T h e first class, cal led super impl ic i t , was deve loped recent ly by the second 0898-1221/03/$ see front matter (~) 2003 Elsevier Science Ltd. All rights reserved. Typeset by ~4~-TEX PII: S0898-1221 (02)00344-9 384 B. NETA AND T. FUKUSHIMA author [2] for the first-order IVPs (1) and for the special second-order IVPs (2). The general form of such methods for the second-order IVPs (2) is given by k g y +l + = h 2 Z # 0 (41 j = l 3=0 For m > 0, the methods are called super implicit because they require the knowledge of functions not only at past and present but also at future t ime steps. } 'hkushima developed Cowell and Adams type super-implicit methods of arbi t rary degree and auxiliary formulae to be used in the s tar t ing procedure. The first step is evaluating Yl using the initial conditions and some future values g 2 (0) Yl = yo + hy~o + h ~ bj fj . (5) j=0 Next, obtain the additional value Y2,. •. , ym-1, using -yn_l + h2 Z b n//j, (6) j = 0 Coefficients b~. n) are given in [2]. In the case of the sixth-order method, we discussed here / 367 3 47 ~ 7 Yl = Yo + hY~o + h 2 ~ l -~ fo + -~fl ~4-6f2 + ouu f3 4 -~f4 , (7) y 2 = 2 y l _ Y o + h 2 ( 1 9 17 7 1 1 ) 2--~ f0 + ~6 f l + 1 -~f2 + ~-6f3 ~-~-~f4 (8) Thus, we have to solve a system of nonlinear equations. In order, to make the system smaller, one can subdivide the total interval of integration to subintervals. This will require special formulae to obta in the ending values. Symmetr ic Cowell type methods of order up to 12 are given along with s tar t ing and ending formulae. The integration error grows linearly with respect to t ime as in symmetr ic mult is tep methods. The second one is due to Obrechkoff 1, see [3]. These methods for the solution of first-order IVPs (1) are given by (see, e.g., [1, pp. 199-204; 4-6]) k g k j=0 i=1 j = 0 According to [6], the error constant decreases more rapidly with increasing g rather than the step k. I t is difficult to satisfy the zero stability for large k. The weak stabili ty interval appears to be small. The advantage of Obrechkoff methods is the fact tha t these are one-step high-order methods and as such do not require additional s tar t ing values. A list of Obrechkoff methods for g = 1 , 2 , . . . , 5 k, k = 1, 2, 3,4 is given in [6]. For example, for k = 1 and g = 2, we get an implicit method of order 4 with an error constant Cs = 1/720, and the method is h h 2 Y n + l Y n ~-~ (Y/n+l qY/n) -~ (Y~+I -Y~) " (10) For k = 1 and g = 3, we get an implicit method of order 6 with an error constant Cr = -1/100800, and the method is h , h 2 ,, h a ,,, y , , ) Y n + l Y n = -~ (Yn+l qY/n) -~ (Yn+l -Y~) q~ (Yn+l -Jr. (11) 1Bulgarian mathemat ic ian Academician Nikola Obrechkoff (1896-1963, born in Varna) who did pioneering work in such diverse fields as analysis, algebra, number theory, numerical analysis, summat ion of divergent series, probabil i ty an statistics. O b r e c h k o f f V e r s u s S u p e r I m p l i c i t M e t h o d s 385 Obrechkoff methods for the solution of second-order IVPs (2) can be found in [7]. Here, P-Stable Obrechkoff methods with minimal phase-lag for periodic initial-value problems are discussed. Also Simos [8] presents P-stable Obrechkoff method. In [9], Obrechkoff methods for general second-order differential equations (3) are developed. Before we continue, we need several definitions. For the multistep method to solve the firstorder IVP k k i = 0 i = 0 we define the characteristic polynomials (see, e.g., [1]) k p(~) = ~ a,~ ~, (13) i = 0 and k ~(~) = ~ b,~'. (14) i = 0 The order of the method is defined to be p if for an adequately smooth arbi t rary test function ¢(x), k k ~ a,¢(x + ih) h ~ b,¢'(x + ih) = cp+lhp+l¢(p+l)(x) + o (hp+2), i=O i=O where Cp+l is the error constant. The method is assumed to satisfy the following: (1) ak = 1, la01 + Ib01 # 0, (2) p and a have no common factor (irreducibility), (3) p(1) = 0, p'(1) = a(1) (consistency), (4) the method is zero-stable (relates to the magnitude of the roots of p). For the multistep method to solve the second-order IVP