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Numerische Mathematik

, Volume 31, Issue 2, pp 131–152

# One-step methods of any order for ordinary differential equations with discontinuous right-hand sides

• Reinhold Mannshardt
Article

## Summary

The right-hand sides of a system of ordinary differential equations may be discontinuous on a certain surface. If a trajectory crossing this surface shall be computed by a one-step method, then a particular numerical analysis is necessary in a neighbourhood of the point of intersection. Such an analysis is presented in this paper. It shows that one can obtain any desired order of convergence if the method has an adequate order of consistency. Moreover, an asymptotic error theory is developed to justify Richardson extrapolation. A general one-step method is constructed satisfying the conditions of the preceding theory. Finally, a simplified Newton iteration scheme is used to implement this method.

## Subject Classifications

AMS(MOS): 65L05 CR: 5.17

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## References

1. 1.
André, J., Seibert, P.: The local theory of piecewise continuous differential equations. In: Contributions to the theory of nonlinear oscillations, Vol. 5 (L. Cesari et al., ed.), pp. 225–255. Princeton: Princeton Univ. Press 1960Google Scholar
2. 2.
De Backer, W.: Jump conditions for sensitivity coefficients. In: Sensitivity methods in control theory (Symp. Dubrovnik 1964; L. Radanović, ed.), pp. 168–175. Oxford: Pergamon Press 1966Google Scholar
3. 3.
Budak, B.M., Gorbunov, A.D.: On the difference method of solution of the Cauchy problem for the equationy′=f(x,y) and for the system of equationsx i=X i(t,x 1,...,x n),i=1,...,n with discontinuous right-hand sides. Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him.5, 7–12 (1958) [in Russian; cf. MR.21 #157 (1960)]Google Scholar
4. 4.
Carver, M.B.: Efficient handling of discontinuities and time delays in ordinary differential equation simulations. In: Proceedings of the international symposium SIMULATION 1977 in Montreux (M.H. Hamza, ed.), pp. 153–158. Anaheim-Calgary-Zürich: Acta Press 1977Google Scholar
5. 5.
Cellier, F.E., Rufer, D.F.: Algorithm suited for the solution of initial value problems in engineering applications. In: Proceedings of the international symposium and course SIMULATION 1975 in Zürich (M.H. Hamza, ed.), pp. 160–165. Calgary-Zürich: Acta Press 1975Google Scholar
6. 6.
Chartres, B., Stepleman, R.: Convergence of difference methods for initial and boundary value problems with discontinuous data. Math. Comput.25, 729–732 (1971)Google Scholar
7. 7.
Chartres, B., Stepleman, R.: A general theory of convergence for numerical methods. SIAM J. Numer. Anal.9, 476–492 (1972)Google Scholar
8. 8.
Chartres, B., Stepleman, R.: Actual order of convergence of Runge-Kutta-methods on differential equations with discontinuities. SIAM J. Numer. Anal.11, 1193–1206 (1974)Google Scholar
9. 9.
Chartres, B., Stepleman, R.: Convergence of linear multistep methods for differential equations with discontinuities. Numer. Math.27, 1–10 (1976)Google Scholar
10. 10.
Evans, D.J., Fatunla, S.O.: Accurate numerical determination of the intersection point of the solution of a differential equation with a given algebraic relation. J. Inst. Math. Appl.16, 355–359 (1975)Google Scholar
11. 11.
Feldstein, A., Goodman, R.: Numerical solution of ordinary and retarded differential equations with discontinuous derivatives. Numer. Math.21, 1–13 (1973)Google Scholar
12. 12.
Filippov, A.F.: Differential equations with discontinuous right-hand side. Mat. Sb. (N.S.)51(93), 99–128 (1960) [in Russian]. Engl. Transl.: AMS Translations, Series 2, Vol. 42, pp. 199–231 (1964)Google Scholar
13. 13.
Halin, H.J.: Integration of ordinary differential equations containing discontinuities. In: Proceedings of the Summer Computer Simulation Conference 1976, pp. 46–53. La Jolla: SCI Press 1976Google Scholar
14. 14.
Hay, J.L.: Object program structures for simulation language translators. In: Proceedings of the symposium on simulation languages for dynamic systems in London 1975, 16/1-7. Bruxelles: AICA 1975Google Scholar
15. 15.
Hay, J.L., Crosbie, R.E., Chaplin, R.I.: Integration routines for systems with discontinuities. Comput. J.17, 275–278 (1974)Google Scholar
16. 16.
Henrici, P.: Discrete variable methods in ordinary differential equations. New York-London-Sydney: Wiley 1962Google Scholar
17. 17.
Magnus, K.: Schwingungen. Stuttgart: Teubner 1961Google Scholar
18. 18.
Mannshardt, R.: Eine Darstellung von Gleitbewegungen längs Unstetigkeitsflächen von Differentialgleichungen mit Sprungfunktionen. Z. Angew. Math. Mech.53, 659–665 (1973)Google Scholar
19. 19.
Mannshardt, R.: Simulation of discontinuous systems by use of Runge-Kutta-methods combined with Newton iteration. In: Proceedings of the international symposium SIMULATION 1977 in Montreux (M.H. Hamza, ed.), pp. 163–167. Anaheim-Calgary-Zürich: Acta Press 1977Google Scholar
20. 20.
Ohashi, T.: On the conditions for convergence of one step methods for ordinary differential equations. TRU Math.6, 59–62 (1970) [cf. Zbl. Math.252, 65054 (1973)]Google Scholar
21. 21.
O'Regan, P.G.: Step size adjustment at discontinuities for fourth order Runge-Kutta methods. Comput. J.13, 401–404 (1970)Google Scholar
22. 22.
Squier, D.P.: One-step methods for ordinary differential equations. Numer. Math.13, 176–179 (1969)Google Scholar
23. 23.
Taubert, K.: Differenzenverfahren für gewöhnliche Anfangswertaufgaben mit unstetiger rechter Seite. In: Numerische Behandlung nichtlinearer Integrodifferential- und Differentialgleichungen (Oberwolfach 1973; R. Ansorge, W. Törnig, eds.), pp. 137–148. Berlin-Heidelberg-New York: Springer 1974Google Scholar
24. 24.
Taubert, K.: Differenzenverfahren für Schwingungen mit trockener und zäher Reibung und für Regelungssysteme. Numer. Math.26, 379–395 (1976)Google Scholar
25. 25.
Zverkina, T.S.: Approximate solution of differential equations with retarded argument and differential equations with discontinuous right-hand sides. Trudy Sem. Teor. Diff. Ur. Otklon. Arg., Univ. Druzby Nar. Patrisa Lumumby1, 76–93 (1962) [in Russian; cf. MR.32#2708 (1966)]Google Scholar
26. 26.
Zypkin, Y.Z., Rutman, R.S.: Sensitivity equations for discontinuous systems. In: Sensitivity methods in control theory (Symp. Dubrovnik 1964; L. Radanović, ed.), pp. 195–196. Oxford: Pergamon Press 1966Google Scholar

## Copyright information

© Springer-Verlag 1978

## Authors and Affiliations

• Reinhold Mannshardt
• 1
1. 1.Rechenzentrum der Ruhr-UniversitätBochumGermany (Fed. Rep.)

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