Abstract
There is a large gap between the theoretical results about iterated defect corrections (IDeC) and practical implementations of IDeC methods for stiff systems. This paper tries to close this gap by providing general principles which are essential for the construction of efficient IDeC codes. Numerical results gathered with one particular IDeC based experimental code furnish evidence of the inherent power of the defect correction concept in the context of stiff systems of ordinary differential equations.
Zusammenfassung
Zwischen den theoretischen Resultaten über die iterierte Defektkorrektur (IDeC) und praktischen Implementierungen von IDeC-Verfahren für steife Differentialgleichungen klafft eine beträchtliche Lücke. Die vorliegende Arbeit versucht diese Lücke zu schließen, indem allgemeine Konstruktionsprinzipien für IDeC-Programme angegeben werden. Die im IDeC-Prinzip potentiell vorhandenen Möglichkeiten kommen in numerischen Resultaten zum Ausdruck, die mit einem Experimentierprogramm zur Lösung steifer Differentialgleichungen gewonnen wurden.
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Bjurel, G., Dahlquist, G., Lindberg, B., Linde, S., Odén, L.: Survey of stiff ordinary differential equations. Report NA 70.11. Dept. of Information Processing, The Royal Institute of Technology, Stockholm, 1970.
Byrne, G. D., Hindmarsh, A. C., Jackson, K. R., Brown, H. G.: Comparative test results for two ODE solvers — EPISODE and GEAR. Report ANL-77-19, Argonne National Laboratory, Argonne, Illinois 1977.
Ehle, B. L.: A comparison of numerical methods for solving certain stiff ordinary differential equation. Report 70, Dept. of Math., University of Victoria, 1972.
Enright, W. H., Hull, T. E., Lindberg, B.: Comparing numerical methods for stiff systems of ODEs. Bit15, 10–48 (1975).
Enright, W. H., Hull, T. E.: Comparing numerical methods for the solution of stiff systems of ODEs arising in chemistry, in: Lapidus, Schiesser [19]. pp. 45–66.
Frank, R., Ueberhuber, C. W.: Iterated defect correction for Runge Kutta methods. Report No. 14/75, Inst. f. Num. Math., Technical University of Vienna, 1975.
Frank, R., Ueberhuber, C. W.: Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations. BIT17, 146–159 (1977).
Frank, R., Ueberhuber, C. W.: Collocation and iterated defect correction in Numerical treatment of differential equations (Bulirsch, R., Grigorieff, R. D., Schröder, J., eds.), Lecture Notes in Mathematics 631, pp. 19–34, Berlin-Heidelberg-New York: Springer 1978.
Frank, R., Ueberhuber, C. W.: Iterated defect correction for differential equations, Part I: Theoretical results. Computing20, 207–228 (1978).
Frank, R., Thiemer, J., Ueberhuber, C. W.: Numerische Quadratur mittels Iterierter Defektkorrektur. Report No. 34/78, Inst. f. Num. Math., Technical University of Vienna, 1978.
Frank, R., Macsek, F., Ueberhuber, C. W.: Some results on the asymptotic error behavior of IDeC-methods. Report No. 37/79, Inst. f. Num. Math., Technical University of Vienna, 1979 (to appear).
Gear, C. W.: Numerical initial value problems in ordinary differential equations. Englewood Cliffs, N. J.: Prentice-Hall 1971.
Gordon, M. K., Shampine, L. F.: Typical problems for stiff differential equations. SIGNUM Newsletter10, 41 (1975).
Hairer, E.: On the order of iterated defect correction. Numer. Math.29, 409–424 (1978).
Hindmarsh, A. C.: GEAR: Ordinary differential equations system solver. Report UCID-30001 (Rev. 3), Lawrence Livermore Laboratory, Livermore, 1974.
Hindmarsh, A. C., Byrne, G. D.: EPISODE: An effective package for the integration of systems of ordinary differential equations. Report UCID-30112 (Rev. 1), Lawrence Livermore Laboratory, Livermore 1977.
Hulme, B. L., Daniel, S. L.: COLODE: A collocation subroutine for ordinary differential equations. Report SAND-74-0380, Sandia Laboratories, Albuquerque 1974.
Krogh, F. T.: On testing a subroutine for the numerical integration of ordinary differential equations. J. ACM20, 545–562 (1973).
Lapidus, L., Schiesser, W. E. (eds.): Numerical methods for differential systems. New York-San Francisco-London: Academic Press 1976.
Prothero, A., Robinson, A.: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comp.28, 145–162 (1974).
Shampine, L. F., Watts, H. A.: Global error estimation for ordinary differential equations. ACM Trans. Math. Software2, 172–186 (1976).
Shampine, L. F., Watts, H. A., Davenport, S. M.: Solving nonstiff ordinary differential equations — the state of the art. SIAM Review18, 376–411 (1976).
Shampine, L. F.: Stability properties of Adams codes. ACM Trans. Math. Software4, 323–329 (1978).
Skelboe, S.: The control of order and steplength for backward differentiation formulas. BIT17, 91–107 (1977).
Stetter, H. J.: Analysis of discretization methods for ordinary differential equations. Berlin-Heidelberg-New York: Springer 1973.
Stetter, H. J.: The defect correction principle and discretization methods. Num. Math.29, 425–443 (1978).
Tendler, J. M., Bickart, T. A., Picel, Z.: A stiffly stable integration process using cyclic composite methods. ACM Trans. math. Software4, 339–368 (1978).
Willoughby, R. A. (ed.): Stiff differential systems. New York-London: Plenum Press 1974.
Zadunaisky, P. E.: On the estimation of errors propagated in the numerical integration of ordinary differential equations. Num. Math.27, 21–39 (1976).
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Ueberhuber, C.W. Implementation of defect correction methods for stiff differential equations. Computing 23, 205–232 (1979). https://doi.org/10.1007/BF02252129
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DOI: https://doi.org/10.1007/BF02252129