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Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting

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The numerical solution of two-point boundary value problems and problems of optimal control by shooting techniques requires integration routines. By solving 15 real-life problems four well-known intergrators are compared relative to reliability, fastness and precision. Hints are given, which routines could be used for a problem.

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References

  1. Aziz, A. K. (ed.): Numerical solutions of boundary value problems for ordinary differential equations. Proceedings of a Symposium held at the University of Maryland, 1974. New York: Academic Press 1975

    Google Scholar 

  2. Bauer, H., Neumann, K.: Berechnung optimaler Steuerungen, Maximumprinzip und dynamische Optimierung. Lecture Notes in Operations Research and Mathematical Systems, Vol. 17. Berlin-Heidelberg-New York: Springer 1969

    Google Scholar 

  3. Broyden, C. G.: A class of methods for solving non-linear simultaneous equations. Math. Comp.19, 577–593 (1965)

    Google Scholar 

  4. Bulirsch, R.: Numerical calculation of elliptic integrals and elliptic functions, I, II. Numer. Math.7, 78–90, 353-354 (1965)

    Google Scholar 

  5. Bulirsch, R., Oettli, W., Stoer, J. (eds.): Optimization and optimal control. Proceedings of a Conference held at Oberwolfach, 1974, Lecture Notes, Vol. 477. Berlin-Heidelberg-New York: Springer 1975

    Google Scholar 

  6. Bulirsch, R., Stoer, J.: Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math.8, 1–13 (1966)

    Google Scholar 

  7. Bulirsch, R., Stoer, J., Deuflhard, P.: Numerical solution of nonlinear two-point boundary value problems, I. Numer. Math. (to be published)

  8. Cole, J. D., Keller, H. B., Saffmann, P. G.: The flow a viscous compressible fluid through a very narrow gap. SIAM J. Appl. Math.15, 605–617 (1967)

    Google Scholar 

  9. Daniel, J. W., Martin, A. J.: Implementing deferred corrections for Numerov's difference method for second-order two-point boundary value problems. Tech. Rep. CNA-107, Center for Numerical Analysis, The University of Texas at Austin (1975)

  10. Davenport, S. M., Shampine, L. F., Watts, H. A.: Comparison of some codes for the initial value problem for ordinary differential equations. In: [1], pp. 349–353 (1975)

    Google Scholar 

  11. Deuflhard, P.: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer. Math.22, 289–315 (1974)

    Google Scholar 

  12. Deuflhard, P.: A relaxation strategy for the modified Newton method. In [5], pp. 59–73 1975

    Google Scholar 

  13. Deuflhard, P., Pesch, H.-J., Rentrop, P.: A modified continuation method for the numerical solution of nonlinear two-point boundary value problems by shooting techniques. Numer. Math.26, 327–343 (1976)

    Google Scholar 

  14. Dickmanns, E. D.: Maximum range, three-dimensional lifting planetary entry. NASA Tech. Rep. R-M 199, Marshall Space Flight Center, Alabama (1972)

    Google Scholar 

  15. Dickmanns, E. D., Pesch, H.-J.: Influence of a reradiative heating constraint on lifting entry trajectories for maximum lateral range. 11th International Symposium on Space Technology and Science, Tokyo, July 1975

  16. Enright, W. H., Bedet, R., Farkas, I., Hull, T. E.: Test results on initial value methods for non-stiff ordinary differential equations. Tech. Rep. No. 68, Department of Computer Science, University of Toronto (1974)

  17. England, R.: Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations. Comp. J.12, 166–170 (1969)

    Google Scholar 

  18. Fehlberg, E.: Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweitenkontrolle. Computing4, 93–106 (1969)

    Google Scholar 

  19. Fehlberg, E.: Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweitenkontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing6, 61–71 (1970)

    Google Scholar 

  20. Fermi, E., Pasta, J. R., Ulam, S.: Studies of nonlinear problems, I. Tech. Rep. No. 1940, Los Alamos (1955)

  21. Gragg, W. B.: On extrapolation algorithms for ordinary initial value problems. SIAM J. Numer. Anal. Ser. B2, 384–403 (1965)

    Google Scholar 

  22. Holt, J. F.: Numerical solution of nonlinear two-point boundary value problems by finite difference methods. Comm. ACM.7, 366–373 (1964)

    Google Scholar 

  23. Hull, T. E.: Numerical solutions of initial value problems for ordinary differential equations. In: [1], pp. 3–26 (1975)

    Google Scholar 

  24. Hussels, H. G.: Schrittweitensteuerung bei der Integration gewöhnlicher Differentialgleichungen mit Extrapolationsverfahren. Universität zu Köln, Mathemathisches Institut, Diplomarbeit (1973)

  25. Keller, H. B.: Numerical methods for two-point boundary value problems. Chapter 6: Practical examples and computational exercises. London: Blaisdell 1968

    Google Scholar 

  26. King, W. S., Lewellen, W. S.: Boundary-layer similarity solutions for rotating flows with and without magnetic interaction. Ref. ATN-63 (9227)-6, Aerodynamics and Propulsion Res. Lab., Aerospace Corp., El Segundo. Calif. (1963)

    Google Scholar 

  27. Küpper, T.: A singular bifurcation problem. Math. Rep. No. 99, Battelle Advanced Studies Center, Geneva (1976)

    Google Scholar 

  28. Lentini, M., Pereyra, V.: Boundary problem solvers for first order systems based on deferred corrections. In: [1], pp. 293–315 (1975)

    Google Scholar 

  29. Na, T. Y., Tang, S. C.: A method for the solution of conduction heat transfer with non-linear heat generation. ZAMM49, 45–52 (1969)

    Google Scholar 

  30. Pereyra, V.: High order finite difference solution of differential equations. Report STAN-CS-73-348, Computer Science Department, Stanford University (1973)

  31. Pesch, H.-J.: Numerische Berechnung optimaler Steuerungen mit Hilfe der Mehrzielmethode dokumentiert am Problem der Rückführung eines Raumgleiters unter Berücksichtigung von Aufheizungsbegrenzungen. Universität zu Köln, Mathematisches Institut, Diplomarbeit (1973)

  32. Pimbley, G.: Eigenfunction branches of nonlinear operators and their bifurcations. Lecture Notes, Vol. 104. Berlin-Heidelberg-New York: Springer 1969

    Google Scholar 

  33. Reissner, E., Weinitschke, H. J.: Finite pure bending of circular cylindrical tubes. Quarterly Appl. Math.20, 305–319 (1963)

    Google Scholar 

  34. Rentrop, P.: Numerical solution of the singular Ginzburg-Landau equations by multiple shooting. Computing16, 61–67 (1976)

    Google Scholar 

  35. Roberts, S. M., Shipman, J. S.: Two-point boundary value problems: Shooting methods. New York: Elsevier 1972

    Google Scholar 

  36. Russel, R. D., Shampine, L. F.: A collocation method for boundary value problems. Numer. Math.19, 1–28 (1972)

    Google Scholar 

  37. Scott, M. R.: Invariant imbedding and its applications to ordinary differential equations: An introduction. Reading, Massachusetts: Addison-Wesley 1973

    Google Scholar 

  38. Scott, M. R., Watts, H. A.: SUPORT—A computer code for two-point boundary-value problems via orthonormalization. Tech. Rep. SAND-75-0198. Sandia Laboratories, Albuquerque, New Mexico (1975)

    Google Scholar 

  39. Sedgwick, A. E.: An effective variable order variable step Adams method. Ph. D. Thesis, Tech. Rep. No. 53, Department of Computer Science, University of Toronto (1973)

  40. Stoer, J., Bulirsch, R.: Einführung in die Numerische Mathematik, 11. Heidelberger Taschenbuch, Bd. 114. Berlin-Heidelberg-New York: Springer 1973

    Google Scholar 

  41. Stoleru, L. G.: An optimal policy for economic growth. Econometrica33, 321–348 (1965)

    Google Scholar 

  42. Thurston, G. A.: Newton's method applied to problems in nonlinear mechanics. J. Appl. Mech.32, 383–388 (1965)

    Google Scholar 

  43. Weinitschke, H. J.: Die Stabilität elliptischer Zylinderschalen bei reiner Biegung. ZAMM50, 411–422 (1970)

    Google Scholar 

  44. Weinitschke, H. J.: On nonsymmetric bending instability of elliptic cylindrical tubes. Unpublished manuscript, TU Berlin (to appear)

  45. Wick, R.: Numerische Lösung volkswirtschaftlicher Variationsprobleme mit Zustandsbeschränkungen unter Anwendung der Mehrzielmethode. Universität zu Köln, Mathematisches Institut, Diplomarbeit (1973)

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Diekhoff, H.J., Lory, P., Oberle, H.J. et al. Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting. Numer. Math. 27, 449–469 (1976). https://doi.org/10.1007/BF01399607

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