Skip to main content
SpringerLink
Log in

Evolutionary Derivation of Sixth-Order P-stable SDIRKN Methods for the Solution of PDEs with the Method of Lines

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

Evolutionary techniques are used for the derivation of a six-stage sixth-order singly diagonally implicit Runge–Kutta–Nyström (SDIRKN) method for the integration of second-order initial value problems (IVPs). This method is P-stable and is recommended for stiff and mildly stiff problems possessing an oscillatory solution. It also attains an order which is one higher than existing methods of this type. Thus, it outperforms the other existing methods of this type when applied to relevant systems of IVPs arising from the semi-disrcetization of partial differential equations (PDEs) with the method of lines (MoL).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Van Der Houwen, P.J., Sommeijer, B.P., Cong, N.H.: Stability of collocation-based Runge–Kutta–Nyström methods. BIT 31, 469–481 (1991)

    Article  MathSciNet  Google Scholar 

  2. Schiesser, W.E., Griffiths, G.W.: A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  3. Bratsos, A.G., Tsitouras, Ch., Natsis, D.G.: Linearized numerical schemes for the Boussinesq equation. Appl. Numer. Anal. Comput. Math. 2, 34–53 (2005)

    Article  MathSciNet  Google Scholar 

  4. Tsitouras, Ch., Famelis, I.T.: Quadratic Störmer-type methods for the solution of the Boussinesq equation by the method of lines. Numer. Methods Partial Differ. Equ. 24, 1321–1328 (2008)

    Article  Google Scholar 

  5. Berg, D.B., Simos, T.E., Tsitouras, Ch.: Trigonometric fitted, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci. 41, 1845–1854 (2018)

    Article  MathSciNet  Google Scholar 

  6. Medvedev, M.A., Simos, T.E., Tsitouras, Ch.: Fitted modifications of Runge-Kutta pairs of orders 6(5). Math. Methods Appl. Sci. 41, 6184–6194 (2018)

    Article  MathSciNet  Google Scholar 

  7. Medvedev, M.A., Simos, T.E., Tsitouras, Ch.: Trigonometric-fitted hybrid four-step methods of sixth order for solving \(y^{\prime \prime }=f(x, y)\). Math. Methods Appl. Sci. 42, 710–716 (2019)

    Article  MathSciNet  Google Scholar 

  8. Medvedev, M.A., Simos, T.E., Tsitouras, Ch.: Hybrid, phase-fitted, four-step methods of seventh order for solving \(x^{\prime \prime }(t) = f (t, x)\). Math. Methods Appl. Sci. 42, 2025–2032 (2019)

    Article  MathSciNet  Google Scholar 

  9. Ramos, H., Kalogiratou, Z., Monovasilis, Th, Simos, T.E.: An optimized two-step hybrid block method for solving general second order initial-value problems. Numer. Algorithms 72, 1089–1102 (2016)

    Article  MathSciNet  Google Scholar 

  10. Simos, T.E.: High order closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation. Appl. Math. Comput. 209, 137–151 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Simos, T.E.: Closed Newton–Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems. Appl. Math. Lett. 22, 1616–1621 (2009)

    Article  MathSciNet  Google Scholar 

  12. Simos, T.E.: New stable closed Newton–Cotes trigonometrically fitted formulae for long-time integration. Abstr. Appl. Anal. 2012 (2012). https://doi.org/10.1155/2012/182536 (Article ID 182536)

    Article  MathSciNet  Google Scholar 

  13. Simos, T.E.: Optimizing a hybrid two-step method for the numerical solution of the Schrödinger equation and related problems with respect to phase-lag. J. Appl. Math. 2012 (2012). https://doi.org/10.1155/2012/420387 (Article ID 420387)

    Article  Google Scholar 

  14. Tsitouras, Ch., Th, I., Famelis, I.T., Simos, T.E.: Phase-fitted Runge-Kutta pairs of orders 8(7). J. Comput. Appl. Math. 321, 226–231 (2017)

    Article  MathSciNet  Google Scholar 

  15. Tsitouras, Ch., Simos, T.E.: Trigonometric fitted explicit Numerov type method with vanishing phase-lag and its first and second derivatives. Mediterr. J. Math. 15, 168 (2018)

    Article  MathSciNet  Google Scholar 

  16. Kalogiratou, Z., Monovasilis, Th, Simos, T.E.: New modified Runge–Kutta–Nystrom methods for the numerical integration of the Schrödinger equation. Comput. Math. Appl. 60, 1639–1647 (2010)

    Article  MathSciNet  Google Scholar 

  17. Kosti, A.A., Anastassi, Z.A., Simos, T.E.: Construction of an optimized explicit Runge–Kutta–Nyström method for the numerical solution of oscillatory initial value problems. Comput. Math. Appl. 61, 3381–3390 (2011)

    Article  MathSciNet  Google Scholar 

  18. Monovasilis, Th, Kalogiratou, Z., Simos, T.E.: Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inf. Sci. 7, 81–85 (2013)

    Article  MathSciNet  Google Scholar 

  19. Monovasilis, T., Kalogiratou, Z., Simos, T.E.: Construction of exponentially fitted symplectic Runge–Kutta–Nyström methods from partitioned Runge–Kutta methods. Mediterr. J. Math. 13, 2271–2285 (2016)

    Article  MathSciNet  Google Scholar 

  20. Papadopoulos, D.F., Simos, T.E.: A modified Runge–Kutta–Nyström method by using phase lag properties for the numerical solution of orbital problems. Appl. Math. Inf. Sci. 7, 433–437 (2013)

    Article  MathSciNet  Google Scholar 

  21. Papadopoulos, D.F., Simos, T.E.: The use of phase lag and amplification error derivatives for the construction of a modified Runge–Kutta–Nyström method. Abstr. Appl. Anal. 2013 (2013) (Article ID: 910624)

  22. Butcher, J.C.: Implicit Runge–Kutta processes. Math. Comput. 18, 50–64 (1964)

    Article  MathSciNet  Google Scholar 

  23. Butcher, J.C.: On Runge–Kutta processes of high order. J. Aust. Math. Soc. 4, 179–194 (1964)

    MathSciNet  MATH  Google Scholar 

  24. Van der Houwen, P.J., Sommeijer, B.P.: Diagonally implicit Runge–Kutta–Nyström methods for oscillatory problems. SIAM J. Numer. Anal. 26, 414–429 (1989)

    Article  MathSciNet  Google Scholar 

  25. Sharp, P.W., Fine, J.M., Burrage, K.: Two-stage and three-stage diagonally implicit Runge–Kutta–Nyström methods of order three and four. IMA J. Numer. Anal. 10, 489–504 (1990)

    Article  MathSciNet  Google Scholar 

  26. Papageorgiou, G., Famelis, I.T., Tsitouras, C.: A P-stable singly diagonally implicit Runge–Kutta–Nyström method. Numer. Algorithms 17, 345–353 (1998)

    Article  MathSciNet  Google Scholar 

  27. Tsitouras, Ch.: Stage reduction on P-stable Numerov type methods of eighth order. J. Comput. Appl. Math. 191, 297–305 (2006)

    Article  MathSciNet  Google Scholar 

  28. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)

    MATH  Google Scholar 

  29. Dong, Ming, Simos, T.E.: A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation. Filomat 31, 4999–5012 (2017)

    Article  MathSciNet  Google Scholar 

  30. Hui, Fei, Simos, T.E.: Four stages symmetric two-step P-stable method with vanished phase-lag and its first, second, third and fourth derivatives. Appl. Comput. Math. 15, 220–238 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Zhang, Wei, Simos, T.E.: A high-order two-step phase-fitted method for the numerical solution of the Schrödinger equation. Mediterr. J. Math. 13, 5177–5194 (2016)

    Article  MathSciNet  Google Scholar 

  32. Panopoulos, G.A., Simos, T.E.: An optimized symmetric 8-step semi-embedded predictor–corrector method for IVPs with oscillating solutions. Appl. Math. Inf. Sci. 7, 73–80 (2013)

    Article  MathSciNet  Google Scholar 

  33. Panopoulos, G.A., Simos, T.E.: A new optimized symmetric embedded predictor–corrector method (EPCM) for initial-value problems with oscillatory solutions. Appl. Math. Inf. Sci. 8, 703–713 (2014)

    Article  Google Scholar 

  34. Panopoulos, G.A., Simos, T.E.: An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown. J. Comput. Appl. Math. 290, 1–15 (2015)

    Article  MathSciNet  Google Scholar 

  35. Papakostas, S.N., Tsitouras, Ch.: High phase-lag order Runge–Kutta and Nyström pairs. SIAM J. Sci. Comput. 21, 747–763 (1999)

    Article  MathSciNet  Google Scholar 

  36. Alolyan, I., Anastassi, Z.A., Simos, T.E.: A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems. Appl. Math. Comput. 218, 5370–5382 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Anastassi, Z.A., Simos, T.E.: A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems. J. Comput. Appl. Math. 236, 3880–3889 (2012)

    Article  MathSciNet  Google Scholar 

  38. Kalogiratou, Z., Monovasilis, Th, Ramos, Higinio, Simos, T.E.: A new approach on the construction of trigonometrically fitted two step hybrid methods. J. Comput. Appl. Math. 303, 146–155 (2016)

    Article  MathSciNet  Google Scholar 

  39. Simos, T.E.: Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation. Acta Appl. Math. 110, 1331–1352 (2010)

    Article  MathSciNet  Google Scholar 

  40. Simos, T.E.: On the explicit four-step methods with vanished phase-lag and its first derivative. Appl. Math. Inf. Sci. 8, 447–458 (2014)

    Article  MathSciNet  Google Scholar 

  41. Simos, T.E.: Multistage symmetric two-step P-stable method with vanished phase-lag and its first, second and third derivatives. Appl. Comput. Math. 14, 296–315 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Stavroyiannis, S., Simos, T.E.: Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs. Appl. Numer. Math. 59, 2467–2474 (2009)

    Article  MathSciNet  Google Scholar 

  43. Simos, T.E., Tsitouras, Ch., Famelis, I.T.: Explicit Numerov type methods with constant coefficients: a review. Appl. Comput. Math. 16, 89–113 (2017)

    MathSciNet  MATH  Google Scholar 

  44. Simos, T.E., Tsitouras, Ch.: A new family of 7 stages, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci. 40, 7867–7878 (2017)

    Article  MathSciNet  Google Scholar 

  45. Tsitouras, Ch., Famelis, I.T.: Symbolic derivation of Runge-Kutta-Nyström order conditions. J. Math. Chem. 46, 896–912 (2009)

    Article  MathSciNet  Google Scholar 

  46. Wolfram Research, Inc., Mathematica, Version 11.1. Wolfram Research, Inc., Champaign (2017)

  47. Famelis, I.T., Papakostas, S.N., Tsitouras, Ch.: Symbolic derivation of Runge-Kutta order conditions. J. Symb. Comput. 37, 311–327 (2004)

    Article  MathSciNet  Google Scholar 

  48. Tsitouras, Ch., Famelis, I.T., Simos, T.E.: On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62, 2101–2111 (2011)

    Article  MathSciNet  Google Scholar 

  49. Famelis, I.T., Tsitouras, Ch.: Symbolic derivation of order conditions for hybrid Numerov-type methods solving \(y^{\prime \prime }=f(x, y)\). J. Comput. Appl. Math. 218, 543–555 (2008)

    Article  MathSciNet  Google Scholar 

  50. Storn, R., Price, K.: Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11, 341–359 (1997)

    Article  MathSciNet  Google Scholar 

  51. Simos, T.E., Tsitouras, Ch.: Evolutionary generation of high order, explicit, two step methods for second order linear IVPs. Math. Meth. Appl. Sci. 40, 6276–6284 (2017)

    Article  MathSciNet  Google Scholar 

  52. Franco, J.M., Gómez, I., Rández, L.: Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order. Numer. Algorithms 26, 347–363 (2001)

    Article  MathSciNet  Google Scholar 

  53. Alonso-Mallo, I., Cano, B., Moreta, M.J.: Stability of Runge–Kutta–Nyström methods. J. Comput. Appl. Maths. 189, 120–131 (2006)

    Article  Google Scholar 

  54. Alolyan, I., Simos, T.E.: A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Comput. Math. Appl. 62, 3756–3774 (2011)

    Article  MathSciNet  Google Scholar 

  55. Simos, T.E., Tsitouras, Ch.: Fitted modifications of classical Runge–Kutta pairs of orders 5(4). Math. Methods Appl. Sci. 41, 4549–4559 (2018). https://doi.org/10.1002/mma.4913

    Article  MathSciNet  MATH  Google Scholar 

  56. Simos, T.E., Tsitouras, Ch.: Fitted modifications of Runge–Kutta pairs of orders 6(5). Math. Methods Appl. Sci. 41, 6184–6194 (2018). https://doi.org/10.1002/mma.5128

    Article  MathSciNet  MATH  Google Scholar 

  57. Tsitouras, Ch.: Optimized explicit Runge–Kutta pair of orders 9(8). Appl. Numer. Math. 38, 123–134 (2001)

    Article  MathSciNet  Google Scholar 

  58. Tsitouras, Ch., Simos, T.E.: On ninth order, explicit Numerov type methods with constant coefficients. Mediterr. J. Math. 15 (2018) (Article No: 46)

Download references

Author information

Authors and Affiliations

  1. School of Art, Ningbo Polytechnic, Ningbo, 315800, China

    Chialiang Lin

  2. Department of Computer Science and Information Engineering, Chang Jung Christian University, Tainan, 71101, Taiwan

    Jwu Jenq Chen

  3. Department of Mathematics, College of Sciences, King Saud University, P. O. Box 2455 , Riyadh, 11451, Saudi Arabia

    T. E. Simos

  4. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

    T. E. Simos

  5. Modern Computational Methods, Ural Federal University, 19 Mira St., 620002, Yekaterinburg, Russian Federation

    T. E. Simos

  6. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece

    T. E. Simos

  7. 10 Konitsis Street, 175 64, Athens, Greece

    T. E. Simos

  8. General Department, National and Kapodistrian University of Athens, Euripus Campus, 34400, Euboea, Greece

    Ch. Tsitouras

Authors
  1. Chialiang Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jwu Jenq Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. E. Simos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ch. Tsitouras
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. E. Simos.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Mathematica verification of the above follows. All computations are valid to quadruple precision.

The coefficients at 32 digits of accuracy.

figure c

Satisfaction of order conditions listed in Table 2.

figure d

Estimation of principal truncation coefficients norms \(\left\| T^{(7)}\right\| _2\,\mathrm{and}\left\| T^{\prime (7)}\right\| _2\).

figure e

Construction of propagation matrix \(R(z)\equiv R(v^2)\).

figure f

Verification that \(P\equiv 1\) over a wide range of values \(v\in [0,20]\).

figure g

Verification that \(|S(v^2)|\le 2\).

figure h

The reader may also verify \(|S(v^2)|\le 2\) by plotting \(S(v^2)\).

figure i

The numerical tests in this paper were performed in double precision which is appropriate for a sixth-order method. Writing now:

figure j

etc., we may also verify that the above are valid for double precision also.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, C., Chen, J.J., Simos, T.E. et al. Evolutionary Derivation of Sixth-Order P-stable SDIRKN Methods for the Solution of PDEs with the Method of Lines. Mediterr. J. Math. 16, 69 (2019). https://doi.org/10.1007/s00009-019-1336-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-019-1336-8

Keywords

Mathematics Subject Classification

Search

Navigation