Skip to main content
SpringerLink
Log in
Book cover

Recent Advances in Computational and Applied Mathematics pp 41–66Cite as

Advances on Collocation Based Numerical Methods for Ordinary Differential Equations and Volterra Integral Equations

  • Conference paper

Abstract

We present a survey on collocation based methods for the numerical integration of Ordinary Differential Equations (ODEs) and Volterra Integral Equations (VIEs), starting from the classical collocation methods, to arrive to the most important modifications appeared in the literature, also considering the multistep case and the usage of basis of functions other than polynomials.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Bartoszewski, Z., Jackiewicz, Z.: Derivation of continuous explicit two-step Runge–Kutta methods of order three. J. Comput. Appl. Math. 205, 764–776 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartoszewski, Z., Podhaisky, H., Weiner, R.: Construction of stiffly accurate two-step Runge–Kutta methods of order three and their continuous extensions using Nordsieck representation. Report No. 01, Martin-Luther-Universität Halle-Wittenberg, Fachbereich Mathematik und Informatik (2007)

    Google Scholar 

  3. Bellen, A., Jackiewicz, Z., Vermiglio, R., Zennaro, M.: Natural continuous extensions of Runge–Kutta methods for Volterra integral equations of the second kind and their applications. Math. Comput. 52(185), 49–63 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellen, A., Jackiewicz, Z., Vermiglio, R., Zennaro, M.: Stability analysis of Runge–Kutta methods for Volterra integral equations of the second kind. IMA J. Numer. Anal. 10(1), 103–118 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bother, P., De Meyer, H., Vanden Berghe, G.: Numerical solution of Volterra equations based on mixed interpolation. Comput. Math. Appl. 27, 1–11 (1994)

    Google Scholar 

  6. Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  7. Brunner, H., van der Houwen, P.J.: The Numerical Solution of Volterra Equations. CWI Monographs, vol. 3. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  8. Brunner, H., Makroglou, A., Miller, R.K.: On mixed collocation methods for Volterra integral equations with periodic solution. Appl. Numer. Math. 24, 115–130 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brunner, H., Makroglou, A., Miller, R.K.: Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution. Appl. Numer. Math. 23, 381–402 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Runge–Kutta and General Linear Methods. Wiley, Chichester/New York (1987)

    MATH  Google Scholar 

  11. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)

    Book  MATH  Google Scholar 

  12. Butcher, J.C., Tracogna, S.: Order conditions for two-step Runge–Kutta methods. Appl. Numer. Math. 24, 351–364 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Capobianco, G., Conte, D., Del Prete, I., Russo, E.: Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations. BIT 47(2), 259–275 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Capobianco, G., Conte, D., Del Prete, I., Russo, E.: Stability analysis of fast numerical methods for Volterra integral equations. Electron. Trans. Numer. Anal. 30, 305–322 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Cardone, A., Ixaru, L.Gr., Paternoster, B.: Exponential fitting Direct Quadrature methods for Volterra integral equations. Numer. Algorithms (2010). doi:10.1007/s11075-010-9365-1

    MathSciNet  Google Scholar 

  16. Chan, R.P.K., Leone, P., Tsai, A.: Order conditions and symmetry for two-step hybrid methods. Int. J. Comput. Math. 81(12), 1519–1536 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Coleman, J.P.: Rational approximations for the cosine function; P-acceptability and order. Numer. Algorithms 3(1–4), 143–158 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Coleman, J.P.: Mixed interpolation methods with arbitrary nodes. J. Comput. Appl. Math. 92, 69–83 (1998)

    Article  MathSciNet  Google Scholar 

  19. Coleman, J.P.: Order conditions for a class of two-step methods for y″=f(x,y). IMA J. Numer. Anal. 23, 197–220 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Coleman, J.P., Duxbury, S.C.: Mixed collocation methods for y″=f(x,y). J. Comput. Appl. Math. 126(1–2), 47–75 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Coleman, J.P., Ixaru, L.Gr.: P-stability and exponential-fitting methods for y″=f(x,y). IMA J. Numer. Anal. 16(2), 179–199 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cong, N.H., Mitsui, T.: Collocation-based two-step Runge–Kutta methods. Jpn. J. Ind. Appl. Math. 13(1), 171–183 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Conte, D., Del Prete, I.: Fast collocation methods for Volterra integral equations of convolution type. J. Comput. Appl. Math. 196(2), 652–663 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Conte, D., Paternoster, B.: A family of multistep collocation methods for Volterra integral equations. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) Numerical Analysis and Applied Mathematics. AIP Conference Proceedings, vol. 936, pp. 128–131. Springer, Berlin (2007)

    Google Scholar 

  25. Conte, D., Paternoster, B.: Multistep collocation methods for Volterra integral equations. Appl. Numer. Math. 59, 1721–1736 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Conte, D., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comput. 204, 839–853 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Conte, D., D’Ambrosio, R., Ferro, M., Paternoster, B.: Piecewise-polynomial approximants for solutions of functional equations, in press on Collana Scientifica di Ateneo, Univ. degli Studi di Salerno

    Google Scholar 

  28. Conte, D., D’Ambrosio, R., Ferro, M., Paternoster, B.: Practical construction of two-step collocation Runge–Kutta methods for ordinary differential equations. In: Applied and Industrial Mathematics in Italy III. Series on Advances in Mathematics for Applied Sciences, pp. 278–288. World Scientific, Singapore (2009)

    Google Scholar 

  29. Conte, D., D’Ambrosio, R., Ferro, M., Paternoster, B.: Modified collocation techniques for Volterra integral equations. In: Applied and Industrial Mathematics in Italy III. Series on Advances in Mathematics for Applied Sciences, pp. 268–277. World Scientific, Singapore (2009)

    Google Scholar 

  30. Crisci, M.R., Russo, E., Vecchio, A.: Stability results for one-step discretized collocation methods in the numerical treatment of Volterra integral equations. Math. Comput. 58(197), 119–134 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. D’Ambrosio, R.: Highly stable multistage numerical methods for functional equations: theory and implementation issues. Bi-Nationally supervised Ph.D. Thesis in Mathematics, University of Salerno, Arizona State University (2010)

    Google Scholar 

  32. D’Ambrosio, R., Jackiewicz, Z.: Continuous two-step Runge–Kutta methods for ordinary differential equations. Numer. Algorithms 54(2), 169–193 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. D’Ambrosio, R., Jackiewicz, Z.: Construction and implementation of highly stable two-step collocation methods, submitted

    Google Scholar 

  34. D’Ambrosio, R., Ferro, M., Paternoster, B.: A general family of two step collocation methods for ordinary differential equations. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) Numerical Analysis and Applied Mathematics. AIP Conference Proceedings, vol. 936, pp. 45–48. Springer, Berlin (2007)

    Google Scholar 

  35. D’Ambrosio, R., Ferro, M., Paternoster, B.: Two-step hybrid collocation methods for y″=f(x,y). Appl. Math. Lett. 22, 1076–1080 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. D’Ambrosio, R., Ferro, M., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for ordinary differential equations. Numer. Algorithms 53(2–3), 195–217 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. D’Ambrosio, R., Ferro, M., Paternoster, B.: Trigonometrically fitted two-step hybrid methods for second order ordinary differential equations with one or two frequencies, to appear on Math. Comput. Simul.

    Google Scholar 

  38. D’Ambrosio, R., Ferro, M., Paternoster, B.: Collocation based two step Runge–Kutta methods for ordinary differential equations. In: Gervasi, O., et al. (eds.) ICCSA 2008. Lecture Notes in Comput. Sci., Part II, vol. 5073, pp. 736–751. Springer, New York (2008)

    Google Scholar 

  39. De Meyer, H., Vanthournout, J., Vanden Berghe, G.: On a new type of mixed interpolation. J. Comput. Appl. Math. 30, 55–69 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  40. Franco, J.M.: A class of explicit two-step hybrid methods for second-order IVPs. J. Comput. Appl. Math. 187(1), 41–57 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math. 3, 381–397 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  42. Guillou, A., Soulé, F.L.: La résolution numérique des problèmes differentiels aux conditions par des méthodes de collocation. RAIRO Anal. Numér. Ser. Rouge R-3, 17–44 (1969)

    Google Scholar 

  43. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II—Stiff and Differential–Algebraic Problems. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (2002)

    Google Scholar 

  44. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration—Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics. Springer, Berlin (2000)

    Google Scholar 

  45. Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I—Nonstiff Problems. Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (2000)

    Google Scholar 

  46. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York (1962)

    MATH  Google Scholar 

  47. Ixaru, L.Gr.: Operations on oscillatory functions. Comput. Phys. Commun. 105, 1–19 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ixaru, L.Gr., Vanden Berghe, G.: Exponential Fitting. Kluwer Academic, Dordrecht (2004)

    MATH  Google Scholar 

  49. Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, Hoboken (2009)

    Book  MATH  Google Scholar 

  50. Jackiewicz, Z., Tracogna, S.: A general class of two-step Runge–Kutta methods for ordinary differential equations. SIAM J. Numer. Anal. 32, 1390–1427 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  51. Jackiewicz, Z., Tracogna, S.: Variable stepsize continuous two-step Runge–Kutta methods for ordinary differential equations. Numer. Algorithms 12, 347–368 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  52. Jackiewicz, Z., Renaut, R., Feldstein, A.: Two-step Runge–Kutta methods. SIAM J. Numer. Anal. 28(4), 1165–1182 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  53. Konguetsof, A., Simos, T.E.: An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems. Numerical methods in physics, chemistry, and engineering. Comput. Math. Appl. 45(1–3), 547–554 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kramarz, L.: Stability of collocation methods for the numerical solution of y″=f(t,y). BIT 20(2), 215–222 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  55. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. Wiley, Chichester (1991)

    MATH  Google Scholar 

  56. Lambert, J.D., Watson, I.A.: Symmetric multistep methods for periodic initial value problems. J. Inst. Math. Appl. 18, 189–202 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  57. Lie, I.: The stability function for multistep collocation methods. Numer. Math. 57(8), 779–787 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  58. Lie, I., Norsett, S.P.: Superconvergence for multistep collocation. Math. Comput. 52(185), 65–79 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  59. Lopez-Fernandez, M., Lubich, C., Schädle, A.: Fast and oblivious convolution quadrature. SIAM J. Sci. Comput. 28, 421–438 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  60. Lopez-Fernandez, M., Lubich, C., Schädle, A.: Adaptive, fast, and oblivious convolution in evolution equations with memory. SIAM J. Sci. Comput. 30, 1015–1037 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  61. Martucci, S., Paternoster, B.: General two step collocation methods for special second order ordinary differential equations. Paper Proceedings of the 17th IMACS World Congress Scientific Computation, Applied Mathematics and Simulation, Paris, July 11–15 (2005)

    Google Scholar 

  62. Martucci, S., Paternoster, B.: Vandermonde-type matrices in two step collocation methods for special second order ordinary differential equations. In: Bubak, M., et al. (eds.) Computational Science, ICCS 2004. Lecture Notes in Comput. Sci., Part IV, vol. 3039, pp. 418–425. Springer, Berlin (2004)

    Google Scholar 

  63. Norsett, S.P.: Collocation and perturbed collocation methods. In: Numerical Analysis, Proc. 8th Biennial Conf., Univ. Dundee, Dundee, 1979. Lecture Notes in Math., vol. 773, pp. 119–132. Springer, Berlin (1980)

    Chapter  Google Scholar 

  64. Norsett, S.P., Wanner, G.: Perturbed collocation and Runge Kutta methods. Numer. Math. 38(2), 193–208 (1981)

    Article  MathSciNet  Google Scholar 

  65. Paternoster, B.: Runge–Kutta(–Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials. Appl. Numer. Math. 28, 401–412 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  66. Paternoster, B.: A phase-fitted collocation-based Runge–Kutta–Nyström methods. Appl. Numer. Math. 35(4), 339–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  67. Paternoster, B.: General two-step Runge–Kutta methods based on algebraic and trigonometric polynomials. Int. J. Appl. Math. 6(4), 347–362 (2001)

    MathSciNet  MATH  Google Scholar 

  68. Paternoster, B.: Two step Runge–Kutta–Nystrom methods for y″=f(x,y) and P-stability. In: Sloot, P.M.A., Tan, C.J.K., Dongarra, J.J., Hoekstra, A.G. (eds.) Computational Science, ICCS 2002. Lecture Notes in Computer Science, Part III, vol. 2331, pp. 459–466. Springer, Amsterdam (2002)

    Google Scholar 

  69. Paternoster, B.: Two step Runge–Kutta–Nystrom methods for oscillatory problems based on mixed polynomials. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Dongarra, J.J., Zomaya, A.Y., Gorbachev, Y.E. (eds.) Computational Science, ICCS 2003. Lecture Notes in Computer Science, Part II, vol. 2658, pp. 131–138. Springer, Berlin/Heidelberg (2003)

    Google Scholar 

  70. Paternoster, B.: Two step Runge–Kutta–Nystrom methods based on algebraic polynomials. Rend. Mat. Appl., Ser. VII 23, 277–288 (2003)

    MathSciNet  MATH  Google Scholar 

  71. Paternoster, B.: Two step Runge–Kutta–Nystrom methods for oscillatory problems based on mixed polynomials. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Dongarra, J.J., Zomaya, A.Y., Gorbachev, Y.E. (eds.) Computational Science, ICCS 2003. Lecture Notes in Computer Science, Part II, vol. 2658, pp. 131–138. Springer, Berlin/Heidelberg (2003)

    Google Scholar 

  72. Paternoster, B.: A general family of two step Runge–Kutta–Nyström methods for y″=f(x,y) based on algebraic polynomials. In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J.J. (eds.) Computational Science, ICCS 2003. Lecture Notes in Computer Science, Part IV, vol. 3994, pp. 700–707. Springer, Berlin (2006)

    Google Scholar 

  73. Petzold, L.R., Jay, L.O., Yen, J.: Numerical solution of highly oscillatory ordinary differential equations. Acta Numer. 6, 437–483 (1997)

    Article  MathSciNet  Google Scholar 

  74. Simos, T.E.: Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. Appl. Math. Lett. 17(5), 601–607 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  75. Van de Vyver, H.: A phase-fitted and amplification-fitted explicit two-step hybrid method for second-order periodic initial value problems. Int. J. Mod. Phys. C 17(5), 663–675 (2006)

    Article  MATH  Google Scholar 

  76. Van de Vyver, H.: Phase-fitted and amplification-fitted two-step hybrid methods for y″=f(x,y). J. Comput. Appl. Math. 209(1), 33–53 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  77. Van den Houwen, P.J., Sommeijer, B.P., Nguyen, H.C.: Stability of collocation-based Runge–Kutta–Nyström methods. BIT 31(3), 469–481 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  78. Wright, K.: Some relationships between implicit Runge–Kutta, collocation and Lanczos τ-methods, and their stability properties. BIT 10, 217–227 (1970)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Salerno, Fisciano, Italy

    Dajana Conte, Raffaele D’Ambrosio & Beatrice Paternoster

Authors
  1. Dajana Conte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raffaele D’Ambrosio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Beatrice Paternoster
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Beatrice Paternoster .

Editor information

Editors and Affiliations

  1. Dpt. Computer Science and Technology, University of Peloponnese, Tripolis, GR-221 00, Greece

    Theodore E. Simos

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media B.V.

About this paper

Cite this paper

Conte, D., D’Ambrosio, R., Paternoster, B. (2011). Advances on Collocation Based Numerical Methods for Ordinary Differential Equations and Volterra Integral Equations. In: Simos, T. (eds) Recent Advances in Computational and Applied Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9981-5_3

Download citation

Publish with us

Policies and ethics

Search

Navigation