Error Control Policy for Initial Value Problems with Discontinuities and Delays

- Journal title : Kyungpook mathematical journal
- Volume 48, Issue 4, 2008, pp.665-684
- Publisher : Department of Mathematics, Kyungpook National University
- DOI : 10.5666/KMJ.2008.48.4.665

Title & Authors

Error Control Policy for Initial Value Problems with Discontinuities and Delays

Khader, Abdul Hadi Alim A.;

Khader, Abdul Hadi Alim A.;

Abstract

Runge-Kutta-Nystrm (RKN) methods provide a popular way to solve the initial value problem (IVP) for a system of ordinary differential equations (ODEs). Users of software are typically asked to specify a tolerance , that indicates in somewhat vague sense, the level of accuracy required. It is clearly important to understand the precise effect of changing , and to derive the strongest possible results about the behaviour of the global error that will not have regular behaviour unless an appropriate stepsize selection formula and standard error control policy are used. Faced with this situation sufficient conditions on an algorithm that guarantee such behaviour for the global error to be asympotatically linear in as , that were first derived by Stetter. Here we extend the analysis to cover a certain class of ODEs with low-order derivative discontinuities, and the class of ODEs with constant delays. We show that standard error control techniques will be successful if discontinuities are handled correctly and delay terms are calculated with sufficient accurate interpolants. It is perhaps surprising that several delay ODE algorithms that have been proposed do not use sufficiently accurate interpolants to guarantee asymptotic proportionality. Our theoretical results are illustrated numerically.

Keywords

Delay ordinary differential equations;discontinuity;global error;interpolation;local error;defect;residual;tolerance proportionality;Runge-Kutta-Nystrm;

Language

English

Cited by

References

1.

K. Abdul Hadi A., The Behaviour of the Global Error in the Numerical Solution of Ordinary and Integro-Differential Equations. Ph.D. Thesis, University of Dundee, 1997.

2.

H. Arndt, Numerical solution of retarded initial value problems with local and global error and stepsize control, Numer. Math., 43(1984), 343-360.

3.

U. M. Ascher, R. M. M. Mattheij and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, NJ, 1988).

4.

A. Bellen, A Runge-Kutta-Nystrom method for Delay Differential Equations, Progress in Scientific Computing, Vol.5, Numerical Boundary Value ODEs, 1985, 271-283.

5.

A. Bellen, One step collocation for delay differential equations, J. Comp. Appl. Math., 10(1984), 275-283.

6.

A. Bellen and M. Zennaro,Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method, Numer. Math., 47(1985), 301-316.

7.

J. R. Cash, A block 6(4) Runge-Kutta formula for nonstiff initial value problems, ACM Trans. Math. Software, 15(1989), 15-28.

8.

P. Chocholaty and L. Slahor, A method to boundary value problems for delay equations, Numer. Math., 33(1979), 69-75.

9.

J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Math., 6(1980), 19-26.

10.

J. R. Dormand and P. J. Prince, Runge-Kutta- Nystrom triples, Comput. Math. Appl., 12-13(1987), 937-949.

11.

J. R. Dormand, M. E. A. El-Mikkawy, and P. J. Prince, Families of embedded Runge-Kutta-Nystrom formulae, IMA J. Numer. Anal., 7(1987), 235-250.

12.

W. H. Enrigh, K. R. Jackson, S. P. Norsett and P. G. Thomsen, Interpolants for Runge-Kutta formulas, ACM Trans. Math. Software, 12(1986), 193-218.

13.

W. H. Enrigh, K. R. Jackson, S. P. Norsett and P. G. Thomsen, Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants, Appl. Math. Comput., 27(1988), 313-335.

14.

W. H. Enrigh and J. D. Pryce, Tow FORTRAN packages for assessing initial value method, ACM Trans. Math. Software, 13(1987), 1-27.

15.

J. M. Fine, A Low Order Runge-Kutta-NystrÄom Methods with Interpolations, Tech. Report 183/85, University of Toronto, Canada, 1985.

16.

C. W. Gear and O. Osterby, Solving ordinary differential equations with discontinuities, ACM Trans. Math. Software, 10(1984), 23-44.

17.

E. Hairer, S. P. Norsett and G. Wanner, Solving Ordinary Differential Equations I (Springer, Berlin, 1987).

18.

M. K. Horn, Developments in High Order Runge-Kutta-Nystrom Formulas, Dissertation, Texas University, Austin, Texas, 1977.

19.

A. V. Kim and V. G. Pimenov, Numerical methods for time-delay systems on the basis of i-smooth analysis, Proceedings of the 15th World Congress on Scientific Computation, Modelling and Applied Mathematics, Vol.1 : Computational Mathematics, pp. 193-196, 1997.

20.

A. V. Kim and V. G. Pimenov, Numerical methods for delay differential equations. Application of i-smooth calculus. (Lecture Notes in Mathematics, Vol. 44). Research Institute of Mathematics- Global Analysis Research Center. Seoul National University, 1999.

21.

A. Marthinsen. Continuous Extensions to Nystrom methods for the explicit solution of second order initial value problems. Technical report, Norwegian institute of Technology, Division of Mathematical Sciences, 1994.

22.

K. W. Neves, Automatic integration of functional differential equations: an approach, ACM Trans. Math. Software, 7(1981), 421-444.

23.

K. W. Neves and A. Feldstein, Characterization of jump discontinuities for state dependent delay differential equations, J. Math. Anal. Appl., 56(1976), 689-707.

24.

K. W. Neves, Control of interpolatory error in retarded differential equations, ACM Trans. Math. Software, 1(1975), 357-368.

25.

H. J. Oberle and H. J. Pesch,Numerical treatment of delay differential equations by Hermite interpolation, Numer. Math., 37(1981), 235-255.

26.

J. Oppelstrup, The RKFHB4 method for delay differential equations in: R. Burlisch, R.D. Grigorieff and J. Schroder, eds., Numerical Treatment of Differential Equations: Proceedings Oberwolfach, 1976, Lecture Notes in Mathematics 631 (Springer, Berlin, 1978), 133-146.

27.

G. Papageorgiou and Ch. Tsitouras, Practical Runge-Kutta-Nystrom methods for the equation y" = f(t; y) with interpolation properties, Dept. of Mathematics Tech. Report., 1/87, National Technical University of Athens, Greece, 1987.

28.

G. Papageorgiou and Ch. Tsitouras, Scaled Runge-Kutta-NystrÄom methods for the second order differential equation y" = f(t; y), Intern. J. of Computer Mathematics, 28(1989), 139-150.

29.

M. G. Roth, Difference methods for stiff delay differential equations, Ph.D. Thesis, Tech. Report UIUCDCS-R-80 ¡ 1012, Department of Computer Science, University of Illinois at Urbana- Champagne, IL (1980).

30.

L. F. Shampine, S. Thompson and J. Kierzenka, Solving Delay Differential Equations with dde23, Mathematics Department, Southern Methodist University, Dallas,TX 75275, May 2; (2002).

31.

L. F. Shampine, Numerical Solution of ODEs, Southern Methodist University, Chapman & Hall, London.

32.

P.W. Sharp and J.M. Fine, ERNY-an explicit Runge-Kutta-Nystrom integrator for second order initial value problems, TR 199/87, Department of Computer Science, University of Toronto, Toronto, 1987.

33.

H. J. Stetter, Considerations concerning a theory for ODE-solvers, in: R. Burlisch, R.D. Grigorieff and J. Schroder, eds., Numerical Treatment of Differential Equations: Proceedings Oberwolfach, 1976, Lecture Notes in Mathematics 631 (Springer, Berlin, 1978), 188-200.

34.

H. J. Stetter, Interpolation and error estimates in Adams PC-codes, SIAM J. Numer. Anal., 16(1979), 311-323.

35.

H. J. Stetter, Tolerance proportionality in ODE-codes, in: R. Marz, ed., Proceedings Second Conference on Numerical Treatment of Differential Equations, Seminarberichte No.32, Humboldt University, Berlin (1980), 109-123; also in : R.D. Skeel, ed., Working Papers for the 1979 SIGNUM Meeting on Numerical Ordinary Differential Equations, Department of Computer Science, University of Illinois at Urbana-Champagne.

36.

D. R. Wille and C.T.H Baker, The propagation of derivative discontinuities in systems of delay differential equations, Numerical Analysis Report 160,University of Manchester, Manchester, England (1988).

37.

D. R. Wille and C.T.H Baker, The tracking of derivative discontinuities in systems of delay differential equations, Numerical Analysis Report 185,University of Manchester, Manchester,England (1990).