Mathematics of Computation

Author(s): Othmar Koch; Ewa B. Weinmüller.
Journal: Math. Comp. 72 (2003), 289-305.
MSC (2000): Primary 65L10
Posted: December 5, 2001
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Abstract: We investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems. Particular attention is paid to the well-posedness of the process. It is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available. Moreover, convergence results for a perturbed Newton iteration are discussed.

1.: U. ASCHER, R. MATTHEIJ, AND R. RUSSELL, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1988. MR 96f:65075 (reprint)
2.: W. AUZINGER, O. KOCH, P. KOFLER, AND E. WEINMÜLLER, The application of shooting to singular boundary value problems, Techn. Rep. Nr. 126/99, Inst. for Appl. Math. and Numer. Anal., Vienna Univ. of Technology, Austria, 1999.
Available at http://fsmat.at/~othmar/research.html.
3.: E. BADRALEXE AND A. FREEMAN, Eigenvalue equation for a general periodic potential and its multipole expansion solution, Phys. Rev. B, 37 (1988), pp. 1067-1084.
4.: L. BAUER, E. REISS, AND H. KELLER, Axisymmetric buckling of hollow spheres and hemispheres, Comm. Pure Appl. Math., 23 (1970), pp. 529-568. MR 43:4335
5.: T. CARR AND T. ERNEUX, Understanding the bifurcation to traveling waves in a class-b laser using a degenerate Ginzburg-Landau equation, Phys. Rev. A, 50 (1994), pp. 4219-4227.
6.: C. CHAN AND Y. HON, A constructive solution for a generalized Thomas-Fermi theory of ionized atoms, Quart. Appl. Math., 45 (1987), pp. 591-599. MR 88j:34034
7.: M. DRMOTA, R. SCHEIDL, H. TROGER, AND E. WEINMÜLLER, On the imperfection sensitivity of complete spherical shells, Comp. Mech., 2 (1987), pp. 63-74.
8.: R. FAZIO, A novel approach to the numerical solution of boundary value problems on infinite intervals, SIAM J. Numer. Anal., 33 (1996), pp. 1473-1483. MR 97e:65069
9.: R. FRANK, The method of Iterated Defect Correction and its application to two-point boundary value problems, Part I, Numer. Math., 25 (1976), pp. 409-419. MR 56:4180a
10.: -, The method of Iterated Defect Correction and its application to two-point boundary value problems, Part II, Numer. Math., 27 (1977), pp. 407-420. MR 56:4180b
11.: R. FRANK AND C. ÜBERHUBER, Iterated Defect Correction for Runge-Kutta methods, Techn. Rep. Nr. 14/75, Inst. for Appl. Math. and Numer. Anal., Vienna Univ. of Technolgy, Austria, 1975.
12.: F. FROMMLET AND E. WEINMÜLLER, Asymptotic error expansions for singular boundary value problems, Math. Models Methods Appl. Sci., 11 (2001), pp. 71-85. CMP 2001:09
13.: M. HERMANN AND D. KAISER, Shooting methods for two-point BVPs with partially separated endconditions, ZAMM, 75 (1995), pp. 651-668. MR 96i:65061
14.: F. DE HOOG AND R. WEISS, Difference methods for boundary value problems with a singularity of the first kind, SIAM J. Numer. Anal., 13 (1976), pp. 775-813. MR 55:13799
15.: -, The application of linear multistep methods to singular initial value problems, Math. Comp., 31 (1977), pp. 676-690.
16.: -, Collocation methods for singular boundary value problems, SIAM J. Numer. Anal., 15 (1978), pp. 198-217. MR 57:8041
17.: -, The application of Runge-Kutta schemes to singular initial value problems, Math. Comp., 44 (1985), pp. 93-103. MR 86h:65100
18.: H. B. KELLER, Numerical Methods for Two-Point Boundary-Value Problems, Blaisdell Publishing Company, Waltham, Mass., 1968. MR 37:6038
19.: -, Numerical Solution of Two Point Boundary Value Problems, no. 24 in Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. MR 55:6868
20.: O. KOCH, P. KOFLER, AND E. WEINMÜLLER, Analysis of singular initial and terminal value problems, Techn. Rep. Nr. 125/99, Inst. for Appl. Math. and Numer. Anal., Vienna Univ. of Technology, Austria, 1999.
Available at http://fsmat.at/~othmar/research.html.
21.: -, The implicit Euler method for the numerical solution of singular initial value problems, Appl. Num. Math., 34 (2000), pp. 231-252. MR 2001f:65079
22.: O. KOCH AND E. WEINMÜLLER, Iterated Defect Correction for the solution of singular initial value problems.
To appear in SIAM J. Numer. Anal.
23.: P. KOFLER, Theorie und numerische Lösung singulärer Anfangswertprobleme gewöhnlicher Differentialgleichungen mit der Singularität erster Art, Ph. D. Thesis, Inst. for Appl. Math. and Numer. Anal., Vienna Univ. of Technology, Austria, 1998.
24.: P. KOSMOL, Methoden zur numerischen Behandlung nichtlinearer Gleichungen und Optimierungsaufgaben, Teubner, Stuttgart, 1989.
25.: P. LANCASTER, Error analysis for the Newton-Raphson method, Numer. Math., 9 (1966), pp. 55-68. MR 35:1208
26.: X. LIU, A note on the Sturmian Theorem for singular boundary value problems, J. Math. Anal. Appl., 237 (1999), pp. 393-403. MR 2000f:34049
27.: R. M¨ARZ AND E. WEINMÜLLER, Solvability of boundary value problems for systems of singular differential-algebraic equations, SIAM J. Math. Anal., 24 (1993), pp. 200-215.
28.: G. MOORE, Computation and parametrization of periodic and connecting orbits, IMA J. Numer. Anal., 15 (1995), pp. 245-263. MR 96a:34087
29.: M. R. OSBORNE, The stabilized march is stable, SIAM J. Numer. Anal., 16 (1979), pp. 923-933. MR 81e:65047
30.: S. PARTER, M. STEIN, AND P. STEIN, On the multiplicity of solutions of a differential equation arising in chemical reactor theory, Techn. Rep. Nr. 194, Dept. Computer Sciences, Univ. of Wisconsin, 1973 = Studies in Appl. Math., 54 (1975), pp. 293-314. MR 56:10026
31.: H. J. STETTER, Numerik für Informatiker: computergerechte numerische Verfahren, Oldenbourg, Wien-München, 1976. MR 54:1531
32.: R. WEISS, The convergence of shooting methods, BIT, 13 (1973), pp. 470-475. MR 48:12856
33.: H. WERNER AND H. ARNDT, Gewöhnliche Differentialgleichungen: eine Einführung in Theorie und Praxis, Springer-Verlag, Berlin-Heidelberg, 1986. MR 88b:34002

Othmar Koch
Affiliation: Department of Applied Mathematics and Numerical Analysis, University of Technology Vienna, Wiedner Hauptstrasse 8--10, A-1040 Vienna, Austria
Email: othmar@fsmat.at

Ewa B. Weinmüller
Affiliation: Department of Applied Mathematics and Numerical Analysis, University of Technology Vienna, Wiedner Hauptstrasse 8--10, A-1040 Vienna, Austria
Email: e.weinmueller@tuwien.ac.at

DOI: 10.1090/S0025-5718-01-01407-7
PII: S 0025-5718(01)01407-7
Received by editor(s): February 10, 2000
Received by editor(s) in revised form: January 3, 2001
Posted: December 5, 2001
Additional Notes: This project was supported by the Austrian Research Fund (FWF) grant P-12507-MAT
Copyright of article: Copyright 2001, American Mathematical Society

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