×

zbMATH — the first resource for mathematics

Detecting and locating a singular point in the numerical solution of IVPs for ODEs. (English. German summary) Zbl 0761.65054
Most of the available numerical methods for solving initial value problems for ordinary differential equations do not give good results in the neighborhood of a singularity. The objective is to recognize, at very little extra cost, when singularities are affecting the performance of a method.
The authors develop a new approach which can be adopted by standard methods and which does not require special action or information from the user. The approach is comprised of two stages. The first one is a preliminary singularity detection stage, which is straightforward while the second stage confirms the existence of a singularity and attempts to approximate its location. There are three alternative techniques for the second stage. In the first two techniques, based on finding some rational approximation, the location of the singularity is approximated by a zero of the corresponding denominator polynomial, whereas the third technique approximates an interval containing the singularities.
The approach can be used with any Runge-Kutta formula with an associated interpolant. The numerical results show that the techniques are effective in detecting and identifying the location of singularities. The approach is relatively inexpensive to implement and could be employed in standard initial value problems software to signal when a problem is better solved by special purpose methods designed to handle singular problems.

MSC:
65L05 Numerical methods for initial value problems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
Software:
CADRE; dverk
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arndt, H.: Lösung von gewöhnlichen Differentialgleichungen mit nichtlinearen Splines. Numer. Math.33, 323–338 (1979). · Zbl 0414.65048 · doi:10.1007/BF01398648
[2] Baker, G. A.: Recursive calculation of pade approximation. In: Morris, G. (ed.) Pade approximant and their applications, pp. 83–91. New York: Academic Press 1973.
[3] Chang, Y. F.: Conduction-diffusion theory of semiconductor junctions. J. Appl. Physics38, 534–544 (1967). · doi:10.1063/1.1709370
[4] Chang, Y. F.: Automatic solution of differential equations. In: Colton, D. L., Gilbert, R. P. (eds.) Constructive and computational methods for differential equations. Berlin, Heidelberg, New York: Springer 1974. (Lecture Notes in Mathematics430, 61–94)
[5] Chang, Y. F., Corliss, G. F.: Three and five term convergence tests. In Proceedings of the Sixth Manitoba Conference on Numerical Mathematics, Congressus NumerantiumXVII, 135–153 (1976).
[6] Chang, Y. F., Corliss, G. F.: Ratio-like and recurrence relation tests for convergence of series. J. Inst. Maths. Applics.25, 349–359 (1980). · Zbl 0445.65076 · doi:10.1093/imamat/25.4.349
[7] Chang, Y. F., Corliss, G. F.: Solving ordinary differential equations using taylor series. ACM Trans. Math. Software8, 114–144 (1982). · Zbl 0503.65046 · doi:10.1145/355993.355995
[8] Corliss, G. F.: On computing darboux type series analyses, Nonlinear Analysis, Theory, Methods and Applications7, 1247–1253 (1983). · Zbl 0562.65011 · doi:10.1016/0362-546X(83)90056-1
[9] De Boor, C.: Cadre: An algorithm for numerical quadrature. In: John R. Rice (ed.) Mathematical software, pp. 417–449. New York: Academic Press, 1971.
[10] Dormand, J. R., Prince, P. J.: Runge-Kutta triples. Comp. Math. Appl.12, 1007–1017 (1986). · Zbl 0618.65059 · doi:10.1016/0898-1221(86)90025-8
[11] Enright, W. H., Jackson, K. R., Norsett, S. P., Thomsen, P. G.: Interpolants for Runge-Kutta. ACM Trans. Math. Soft.12, 193–218 (1986). · Zbl 0617.65068 · doi:10.1145/7921.7923
[12] Fatunla, S. O.: Numerical treatment of singular IVPs. Comp. Math. Appl.12, 1109–1115 (1986). · Zbl 0659.65071 · doi:10.1016/0898-1221(86)90235-X
[13] Gilewich, J.: Numerical detection of the best Pade approximant and determination of the Fourier coefficients of insufficiently sampled functions. In: Morris, G. (ed.) Pade approximation and their application, pp. 99–103. New York: Academic Press 1973.
[14] Gladwell, I., Shampine, L. F., Baca, L. S., Brankin, R. W.: Practical aspects of interpolation in Runge-Kutta codes. SIAM J. Sci. Stat. Comput.8, 322–341 (1987). · Zbl 0621.65067 · doi:10.1137/0908038
[15] Horn, M. K.: Fourth and fifth order, scaled Runge-Kutta algorithm for treating dense output. SIAM J. Numer. Anal.20, 558–568 (1980). · Zbl 0511.65048 · doi:10.1137/0720036
[16] Hull, T. E., Enright, W. H., Jackson, K. R.,: User’s guide for DVERK A subroutine for solving nonstiff ODE’s, Technical Report No. 100/76, Department of Computer Science, University of Toronto, Canada, 1976.
[17] Hunter, C., Guerrieri, B.: Deducing the properties of singularities of functions from their Taylor series coefficients. SIAM J. Appl. Math.39, 248–263 (1980). · Zbl 0457.65003 · doi:10.1137/0139022
[18] Lambert, J. D., Shaw, B.: On the numerical solution ofy’=f(x,y) by a class of formula based on rational approximation. Math. Comput.19, 456–462 (1965). · Zbl 0131.14402
[19] Lambert, J. D., Shaw, B.: A method for the numerical solution ofy’=f(x,y) based on a self-adjusting non-polynomial interpolant. Math. Comput.20, 11–20 (1966). · Zbl 0133.38205
[20] Luke, Y. L., Fair, W., Wimp, J.: Predictor-corrector formulas based on rational interpolants. Comp. Math. Appl.1, 3–12 (1975). · Zbl 0329.65049 · doi:10.1016/0898-1221(75)90003-6
[21] Lynch, R. E.: Generalized trapezoid formulas and errors in Romberg quadrature, Blanch anniversary volume, Aerospace Research Laboratories, Office of Aerospace Research, United States Air Force, pp. 215–229 (1967).
[22] Lyness, J. N., Ninham, B. W.: Numerical quadrature and asymptotic expansions. Math. Comput.21, 162–178 (1967). · Zbl 0178.18402 · doi:10.1090/S0025-5718-1967-0225488-X
[23] Rall, L. B.: Automatic differentiation: Techniques and applications, Berlin, Heidelberg, New York, Tokyo: Springer 1981. (Lecture Notes in Computer Science, 120) · Zbl 0473.68025
[24] Shampine, L. F.: Interpolation for Runge-Kutta methods. SIAM J. Numer. Anal.22, 1014–1027 (1985). · Zbl 0592.65041 · doi:10.1137/0722060
[25] Shampine, L. F. (1986) Some practical Runge-Kutta formulas. Math. Comp.46, 135–150 (1986). · Zbl 0594.65046 · doi:10.1090/S0025-5718-1986-0815836-3
[26] Suhartanto, H.: A new approach for detecting singular points in the Numerical Solution of Initial Value Problems, M.Sc. Thesis, Department of Computer Science, University of Toronto, Canada, 1990.
[27] Voght, W.: Numerische Verfahren für Anfangswertaufgaben, deren Lösungen Singularitäten besitzen, in Numerisch Behandlung von Differential-gleichungen IV, Wissenschaftliche Beiträge der Friedrich-Schiller Universität Jena, Jena, Germany, pp. 127–144 (1987).
[28] Werner, H.: Calculations of singularities for solutions of algebraic differential equations. In: Werner, H. et al. (eds) Computational aspects of complex analysis, pp. 325–360. Dordrecht: Reidel 1983.
[29] Willers, I. M.: A new integration algorithm for ordinary differential equations based on continued fraction approximation. Comm. ACM17, 504–508 (1974). · Zbl 0285.65045 · doi:10.1145/361147.361150
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.