Highly Stable Super-Implicit Hybrid Methods for Special Second Order IVPs

The idea of symmetric super-implicit linear multi-step methods (SSILMMs) necessitates the use of not just past and present solution values of the ordinary differential equations (ODEs), but also, future values of the solution. Such methods have been proposed recently for the numerical solution of second-order ODEs. One technique to obtain more accurate integration process is to construct linear multi-step methods with hybrid points employing future solution values. In this regard, we construct families of Stӧrmer-Cowell type hybrid SSILMMs having higher order than that of the symmetric super-implicit method recently proposed for the same step number using the Taylors series approach. The newly derived hybrid SSILMMs are p-stable with accurate results when tested on some special second order IVPs.


Introduction
Consider the initial value problem (IVP), = , ; = , = , (1) in ordinary differential equations (ODEs) in which there is no explicit first derivative appearing. There is vast literature for the numerical solution of (1), see [13], [5], and references therein. The linear multi-step methods for solving the second order IVP (1) is, The first and second characteristic polynomials are, = ∑ , = ∑ .
Definition 3 [18]: Method (2) is almost p-stable, if its interval of periodicity is (0,∞) -d, where d is a set of distinct points.
The result put forward by [13] have shown that no LMM (2) of order greater than p = 2 can be p-stable. Also, [8] has proved to support [13]'s claim. Precisely, [8]'s result is stated.
Theorem 4 [8]: Consider an irreducible, convergent, symmetric multi-step method (2). Define the function, Then, the method (2) has a non-vanishing interval of periodicity if and only if, (1) J I has a non-zero double roots in the interval I ∊ 0, P , (2) J′′ I is positive on all the non-zero double roots of J I in interval I ∊ 0, P .
Cash [1] independently showed that the order barrier on attainable order of a p-stable LMM (2) could be bypassed by considering certain hybrid two-step methods. An example of the method from this family is given by, The order is p = 4, % Z = _.
As [1] further noted, [12] claimed to have derived high order p-stable linear multi-step methods but their concept of p-stability is considerably different from that given in [14]. The work in [2] further stressed on the work in [1], by considering the free parameters available in their proposed linear multi-step methods which can reduce the work to two functional evaluations, and also, reduces the work with respect to implementation for nonlinear problems of (1). Fatunla [4] derived a one-leg scheme found to be advantageous in terms of functions evaluations. Only one function evaluation and k values of y need to be stored for use in the next integration step. Fatunla et al [6] used the concept of Padé approximation to obtain a p-stable linear multi-step method, with order p = 4, which is an extension of the scheme in [4]. The concept of p-stability based on [14] (definition (2)) which was also employed in [1] and [6] will be adopted in this paper. Several methods based on LMM have been proposed see for example, [21], [22], [23], and [25]. Neta [16] considered a very special class of (2), the symmetric super-implicit linear multi-step method given by, The are arbitrarily chosen to satisfy zerostability condition, and are the coefficients to be determined. This and like the methods to be proposed require additional formulas to handle the additional starting and future values. The method (11) is the extension of the work in [8]. Example of method (11) derived in [16] is given for k = 4, 9 = 8, However, the Taylors series approach in the sense of the work in [1] will be used to derive the new hybrid extension of (11) in [16] while using MATHEMATICA v 8 [11].

Construction of Hybrid Symmetric Super-Implicit Obreckoff Type LMM
The class of methods to be considered is in the general class of the hybrid method, This is an Obreckoff type class of methods, where the hybrids are given by, In particular, is the hybrid SSILMM, when m = 1, q = 1 in (13), this is also considered in [15], where k and the super-implicit parameter s are even. The method (16) is explicit for s = k -1, implicit for s = k, and super-implicit for s > k with λ ∊ [0, 1] as in [1]. Here the b are fixed, say b 3 = 1, b = -1 to satisfy the zero-stability condition. The constants  (14) and (15), we have, However, the hybrid of interest is where, k and the super-implicit parameter s are even, and λ ∊ [0, 1] as in [1].

Construction of High Order Stӧrmer-Cowell Type Hybrid SSILMMs
This section presents the Stӧrmer-Cowell type hybrid SSILMMs of order p = 10, and p = 12 respectively with hybrid parameter λ. When k = 2, s = 6, and b are arbitrarily chosen as in section (2) and substituted into (16), we have, The following consistent simultaneous order condition are obtained as On the examination of the stability of the new hybrid SSILMM (22), the value of the hybrid parameter n has been carefully chosen as 3 to ensure p-stability. On substituting the hybrid pair (24) and (26) into the main method (22) and applying the scalar test problem (6) for H = :ℎ using MATHEMATICA v 8 [11], we obtain the interval of periodicity (0, ∞). The method is thus p-stable. When k = 2, s = 8, we obtain the order p = 12 method, 3

Implementation of Hybrid SSILMM
Consider the implementation of the new hybrid methods derived to show the accuracy of these methods in solving some stiff oscillatory and undamped Duffing problems of (1) by resolving the problem of implicitness in the derived hybrid methods. However, methods (22) and (27) are consider for implementation following the ideas in [1] and [6]. Assume that (1) is Lipschitz continuous with reference to where L is the Lipschitz constant. The approach of Newton-Raphson iterative method is used to resolve the implicitness in the newly proposed methods. The predictor, of order p = 2 will be used as the starter for the Newton- . The p-stable method, is also employ to generate the future solution values { ± } ,( in the case of (22) and { ± } ƒ ,(,S in the case of (27) respectively. So that the Newton-Raphson iteration becomes where the Jacobian is given by, " The numerical methods (22) and (27) is applied to solve example 1, 2, 3. In the case of the p-stable method in (22), Example 1: Orbital problem (Source: [1], [4], [6], [14]) outward such that its distance from the origin at any given time is given by, The interval 0 < ≤ 40P correspond to 20 orbits of the point , The numerical result is generated using the step size ℎ = -/ , , = 3 1 13, and can be seen in Table 1, 2, and 3.   (41) Where the oscillatory pattern of (40) is generated through the theoretical and numerical solution as in figures 1 and 2 respectively with step size at = 10P.  Example 3: Undamped Duffing IVP (Source: [2], [17], [24]), forced by a harmonic function, with the values of the parameters δ = 0.002 and µ = 1.01, and with the initial conditions 0 = Ÿ, 0 = 0, taking for A the value of the Galerkin approximation at x = 0. By Urabe's method applied to Galerkin's procedure, [20] has computed the Galerkin's approximation of order p = 9 to a periodic solution having the same period as the forcing term with a precision 10 23 of the coefficients of, Where the oscillatory pattern of (42) is generated through the theoretical and numerical solution as in figures 3 and 4 respectively with step size ℎ = at = 40P.

Conclusion
This paper has considered the class of methods defined in  (16), and p-stable methods based on (16) have been derived. In particular, in (16), p-stable methods were derived with order as high as p = 10, and 12 which turns to be higher than that of the ones proposed in [16] for the same step length. The order barrier theorem of [3] which has been extended to second-order ODEs by [9] has been bypassed through the use of hybrid methods. The numerical results compare favourably with theoretical and existing results, see tables 1, 2, 3, and figures 1, 2, 3, and 4 as well.