BIT Numerical Mathematics

, Volume 57, Issue 4, pp 1153–1184 | Cite as

On singular BVPs with nonsmooth data: convergence of the collocation schemes

  • Jana Burkotová
  • Irena Rachůnková
  • Ewa B. Weinmüller


This paper deals with the collocation method applied to solve systems of singular linear ordinary differential equations with variable coefficient matrices and nonsmooth inhomogeneities. The classical stage convergence order is shown to hold for the piecewise polynomial collocation applied to boundary value problems with time singularities of the first kind provided that their solutions are appropriately smooth. The convergence theory is illustrated by numerical examples.


Linear systems of ordinary differential equations Singular boundary value problems Time singularity of the first kind Nonsmooth inhomogeneity Collocation method Convergence 

Mathematics Subject Classification

65L05 65L10 65L20 65L60 



We wish to thank M.Sc. Michael Hubner and M.Sc. Stefan Wurm, Vienna University of Technology, for the numerical simulations.


  1. 1.
    Ascher, U., Bader, A.: A new basis implementation for a mixed order boundary value ODE solver. SIAM J. Sci. Stat. Comput. 8, 483–500 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ascher, U., Christiansen, J., Russell, R.D.: A collocation solver for mixed order systems of boundary values problems. Math. Comp. 33, 659–679 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Auzinger, W., Kneisl, G., Koch, O., Weinmüller, E.B.: A collocation code for boundary value problems in ordinary differential equations. Numer. Algorithms 33, 27–39 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Beletsky, V.V.: Some stability problems in applied mechanics. Appl. Math. Comput. 70, 117–141 (1995)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Buchner, Ch., Schneider, W.: Explosive Crystallization in Thin Amorphous Layers on Heat Conducting Substrates. Poster: 14th International Heat Transfer Conference, Washington, DC, USA, Paper ID IHTC14-22187 (2010)Google Scholar
  6. 6.
    Budd, Ch., Koch, O., Weinmüller, E.B.: Computation of self-similar solution profiles for the nonlinear Schrödinger equation. Computing 77, 335–346 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Burkotová, J., Rachůnková, I., Staněk, S., Weinmüller, E.B.: On linear ODEs with a time singularity of the first kind and nonsmooth inhomogeneity. Bound. Value Probl. 2014, 183 (2014)CrossRefzbMATHGoogle Scholar
  8. 8.
    Burkotová, J., Rachůnková, I., Weinmüller, E.B.: On singular BVPs with nonsmooth data. Analysis of the linear case with variable coefficient matrix. Appl. Numer. Math. 144, 77–96 (2017). doi: 10.1016/j.apnum.2016.06.007
  9. 9.
    Burrage, K., Jackiewicz, Z., Nørsett, S.P., Renaut, R.A.: Preconditioning waveform relaxation iterations for differential systems. BIT 36, 54–76 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    de Boor, C., Swartz, B.: Collocation at Gaussian points. SIAM J. Numer. Anal. 10, 582–606 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    de Hoog, F., Weiss, R.: Collocation methods for singular BVPs. SIAM J. Numer. Anal. 15, 198–217 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    de Hoog, F., Weiss, R.: An approximation theory for boundary value problems on infinite intervals. Computing 24, 227–239 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    de Hoog, F., Weiss, R.: The application of Runge–Kutta schemes to singular initial value problems. Math. Comput. 44, 93–103 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fazio, R.: Free boundary formulation for BVPs on semi-infitite interval and non-iterative transformation methods. Acta Appl. Math. 140, 27–42 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Goldstein, S.: Modern Develpment in Fluid Dynamics. Clarendon, Oxford (1938)Google Scholar
  16. 16.
    Horwath, L.: On the solution of laminar boundary layer equations. Proc. R. Soc. Lond. A 164, 547–579 (1938)CrossRefGoogle Scholar
  17. 17.
    Kierzenka, J., Shampine, L.: A BVP solver that controls residual and error. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 27–41 (2008)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Kitzhofer, G., Koch, O., Weinmüller, E.B.: Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg–Landau equation. BIT Numer. Math. 49, 217–245 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Kitzhofer, G., Koch, O., Pulverer, G., Simon, C., Weinmüller, E.B.: Numerical treatment of singular BVPs: the new matlab code bvpsuite. JNAIAM J. Numer. Anal. Ind. Appl. Math. 5, 113–134 (2010)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Koch, O.: Asymptotically correct error estimation for collocation methods applied to singular boundary value problems. Numer. Math. 101, 143–164 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Koch, O., Weinmüller, E.B.: Analytical and numerical treatment of a singular initial value problem in avalanche modeling. Appl. Math. Comput. 148, 561–570 (2004)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Köppl, A.: Anwendung von Ratengleichungen auf anisotherme Kristallisation von Kunststoffen, Ph. D. Thesis, Vienna Univ. of Technology, Austria (1990)Google Scholar
  23. 23.
    Köppl, A., Berger, J., Schneider, W.: Ausbreitungsgeschwindigkeit und Struktur von Kristallisationswellen. In: Proceedings of GAMM, Stuttgard, Germany (1987)Google Scholar
  24. 24.
    Krauskopf, B., Osinga, H.M., Galn-Vioque, J. (ed.): Numerical Continuation Methods for Dynamical Systems: Path following and boundary value problems. ISBN: (Print) 978-1-4020-6355-8, (Online) 978-1-4020-6356-5 (2007)Google Scholar
  25. 25.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations, Analysis and Numerical Solution. EMS Textbooks in Mathematics. ISBN 978-3-03719-017-3. (2006). doi: 10.4171/017
  26. 26.
    Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Springer Verlag, ISBN: (Print) 978-3-642-27554-8, (Online) 978-3-642-27555-5 (2013)Google Scholar
  27. 27.
    Lentini, M., Keller, H.B.: The von Karman swirling flows. SIAM J. Appl. Math. 38, 52–64 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Lentini, M., Keller, H.B.: Bounadry value problems on semi-infinite intervals and their numerical solutions. SIAM J. Numer. Anal. 17, 577–604 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Link, A., Taubner, A., Wabinski, W., Bruns, T., Elster, C.: Modelling accelerometers for transient signals using calibration measurements upon sinusoidal excitation. Measurement 40, 928–935 (2007)CrossRefGoogle Scholar
  30. 30.
    Markowich, P.A.: Analysis of boundary value problems on infinite intervals. SIAM J. Appl. Math. 14, 11–37 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    McClung, D.M., Hungr, O.: An equation for calculating snow avalanche run- up against barriers. Avalanche Formation, Movement and Effects. IAHS Publications 162, 605–612 (1987)Google Scholar
  32. 32.
    McClung, D.M., Mears, A.I.: Dry-flowing avalanche run-up and run-out. J. Glaciol. 41(138), 359–369 (1995)CrossRefGoogle Scholar
  33. 33.
    Robnik, M.: Matric treatment of wave propagation in stratified media. J. Phys. A Math. Gen. 12(1), 151–158 (1979)CrossRefzbMATHGoogle Scholar
  34. 34.
    Shampine, K., Kierzenka, J., Reichelt, M.: Solving Boundary Value Problems for Ordinary Differential Equations in Matlab with bvp4c (2000)
  35. 35.
    Shampine, L., Muir, P., Xu, H.: User-friendly Fortran BVP solver. JNAIAM J. Numer. Anal. Ind. Appl. Math. 1, 201–217 (2006)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Vainikko, G.: A smooth solution to a linear systems of singular ODEs. J. Anal. Appl. 32, 349–370 (2013)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Vainikko, G.: A smooth solution to a nonlinear systems of singular ODEs. AIP Conf. Proc. 1558, 758–761 (2013)CrossRefGoogle Scholar
  38. 38.
    Weinmüller, E.B.: Collocation for singular boundary value problems of second order. SIAM J. Numer. Anal. 23, 1062–1095 (1986)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Jana Burkotová
    • 1
  • Irena Rachůnková
    • 1
  • Ewa B. Weinmüller
    • 2
  1. 1.Department of Mathematical Analysis and Applications of Mathematics, Faculty of SciencePalacký University OlomoucOlomoucCzech Republic
  2. 2.Department for Analysis and Scientific ComputingVienna University of TechnologyWienAustria

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