Full Length Research Paper
ABSTRACT
A new block method of order five for the numerical solution of initial value problems is derived. The coefficients of the matrix of the method are chosen such that low power of the blocksize appears in the principal local truncation error. The stability polynomial is shown to be a perturbation of order explicit Runge-Kutta method.
Key words: Predictor-corrector methods, ordinary differential equations, block methods.
INTRODUCTION
Numerical methods for the solution of the initial value problems (IVP’s) are well established techniques in literature. In the last two decades, a number of papers have appeared on this topic (Abbas, 1995, 1997; Abbas and Delves, 1989; Voss and Abbas, 1997; Birta and Abou-Rabia, 1987; Chu and Hamilton, 1987; Enright and Pryce, 1987; Franklin, 1978; Henrici, 1962; Houwen and Sommeijer, 1989; Katz et al., 1977; Lambert, 1973; Luther and Konen, 1965; Miranker, 1971; Miranker and Liniger, 1967; Moody and Hans, 1987; Moulton, 1926; Mouney et al., 1991; Rosser, 1967; Shampine and Watts, 1969; Watts and Shampine, 1972; Worland, 1976; Adesanya et al., 2013; Dhaigude and Devkate, 2017; Adesanya et al., 2013). The earliest research on block methods was proposed by Shampine and Watts (1969) and Watts and Shampine (1972) with block implicit one step methods, Chu and Hamilton (1987) with multi-block methods, and Voss and Abbas (1997) with predictor-corrector schemes. Other block methods are discussed by several researches such as Houwen and Sommeijer (1989) with block Runge-Kutta methods. One of such techniques is the block method which by means of a single application of a calculation unit yields a sequence of new estimates for y in the differential equation:
where . If the block size k ≥ 1, then in simple cases the values of x for which solutions are computed will be evenly separated. In other words, each basic cycle of calculation has the potential to advance the solution by k new points in the x direction. The block method of the present study has the properties of Runge-Kutta method for being self-starting and does not refuse the development of separate predictors or starting values. Block method was found to be cost effective and to give better approximation.
DERIVATION OF A FIFTH ORDER METHOD
The requirement that the method be fifth order for any block length k complicates the search for good methods. The aim was to produce a fifth order method which might be very efficient to implement. As far as the order of the accuracy and the stability are concerned, if we require a method of order to be achieved then the local truncation error will normally be of order
which is a very high dependence upon the block length k. We seek some methods where the local truncation errors have a lower power of k, as example,
, but then over experience with order 2 methods suggest that the stability problem will be possibly serious.
Therefore, we consider the fifth order method:
COMPUTATIONAL RESULTS
Tables 1 to 4 show the experimental results for the aforementioned test problems and the accuracy of the solution which is measured by computing the differences between the approximation solution and the exact solution at the end of the steps.
For all the problems tested, the numerical results show that new block fifth order method gives good accuracy with the reduction of total steps.
COMPARISON WITH A RUNGE-KUTTA METHOD OF THE SAME ORDER
Here, we compare the experimental results which was obtained by this study block method (with block length =
5) and the following fifth order Runge-Kutta method given by Luther and Konen (1965). From Tables 5 to 8, we notice that the accuracy is satisfactory and much better than the fifth order Runge-Kutta method. Note that the accuracy of the solution is measured by evaluating (l∞) norm of the differences between the approximation solution and the exact solution.
CONCLUSION
This paper provides a new block method of higher order which can be implemented as a corrected formula of any block size together with Euler’s method as a predictor formula. However, this research is concerned with solving initial value problems using block predictor-corrector method of fifth order. The study highlights the importance of solving initial value problems and their application in the real life problems. Furthermore, the result indicates that the block method which was derived in the study could be considered as a good method on the basis of the accuracy and the comparison with a Runge-Kutta method of the same order. Also, we noticed that the accuracy is satisfactory and much better than the 5th order Runge-Kutta method. Thus, for small h we can say that the algorithm may give a better accuracy. Therefore, the results are acceptable.
CONFLICT OF INTERESTS
The authors have not declared any conflict of interests.
ACKNOWLEDGEMENT
The authors are grateful to the referees for the very constructive comments and for suggesting various improvements to the manuscript.
REFERENCES
Abbas S (1995). A fourth order parallel block predictor-corrector method for IVPs Western Illinois University, USA. |
|
Abbas S, Delves LM (1989). Parallel solution of ODEs by one-step block methods. Report CSMR, University of Liverpool. |
|
Abbas S (1997). Derivation of new block methods for the numerical solution of first-order IVPs.International Journal of Computer Mathematics (64):235-244. |
|
Adesanya AO, Odekunle MR, Alkali MA, Abubakar AB (2013). Start- ing the five steps Stomer-Cowell method by Adams-Bashforth method for the solution of the first order ordinary differential equations. African Journal of Mathematics and Computer Science Research 6(5):89-93. |
|
Adesanya AO, Udoh DM, Ajileye AM (2013). A new hybrid block method for the solution of general third order initial value problems of ordinary dif- ferential equations. International Journal of Pure and Applied Mathematics 86(2):365-375. |
|
Birta LG, Abou-Rabia O (1987). Parallel block predictor Corrector Methods for Ode. IEEE Transactions on Computers (3):299-311. |
|
Chu MT, Hamilton H (1987). Parallel solution of ODEs by multi-block methods. SIAM, Journal on Scientific Computing 3:342-353. |
|
Dhaigude RM, Devkate RK (2017). Solution of first order initial value prob- lem by sixth order predictor corrector method. Global Journal of Pure and Applied Mathematics 13(6):2277-2290. |
|
Enright WH, Pryce JD (1987). Two fortran packages for assessing initial value methods. ACM Transcation on Mathematical Soft-ware. 13(1):1-27. |
|
Franklin MA (1978). Parallel solution of ordinary differential equations. IEEE Trans. Computers (27)C:413-420. |
|
Henrici P (1962). Discrete variable methods in ordinary differential equations. New York: John Wiley and Sons. |
|
Houwen PJ, Sommeijer BJ (1989). Block Runge-Kutta methods on parallel computers. Report NM-R8906. Center for Mathermatic and Computer Science, Amsterdam. Journal Kuwait University 22(2):5-11. |
|
Katz IN, Franklin MA, Sen A (1977). Optimally stable parallel predictors for Adams-Moulton correctors. Computers and Mathematics with Applications 3(3):217-233. |
|
Lambert D (1973). Computational methods in ordinary differential equations. New York: Wiley. |
|
Luther HA, Konen HP (1965). Some fifth-order classical Runge-Kutta formulas. SIAM Review 4(7):551-558. |
|
Miranker WL, Liniger W (1967). Parallel methods for the numerical integration of ordinary differential equations. Math. Mathematics of Computation 21(99):303-320. |
|
Miranker WL (1971). A survey of parallelism in numerical analysis. SIAM Review (13):524-547. |
|
Moody TC, Hans H (1987). Parallel solution of ODEs by multi-block methods. SIAM journal on scientific and statistical computing 8(3):342-353. |
|
Moulton FR (1926). New methods in exterior ballistics. University of Chicago. |
|
Mouney G, Authie G, Gayraud, T (1991). Parallel solution of ODEs. Implementation of block methods on transputer networks, Contributed paper at ICIAM 91, Washington D.C. Press, Chicago. |
|
Rosser JB (1967). A Runge-Kutta for all seasons. SIAM Review (9):417-452. |
|
Shampine LF, Watts HA (1969). Block implicit one-step methods. Mathematics of Computation 23(108):731-740. |
|
Voss D, Abbas S (1997). Block predictor-corrector schemes for the parallel solution of ODEs. Computers and Mathematics with Applications 33(6):65-72. |
|
Watts HA, Shampine LF (1972). A stable block implicit methods. BIT Numerical Mathematics 12(2):252-266. |
|
Worland PB (1976). Parallel methods for the numerical solution of ordinary dif- ferential equations. IEEE Transactions on Computers 10:1045-1048. |
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