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

  • Chialiang Lin
  • Jwu Jenq Chen
  • T. E. SimosEmail author
  • Ch. Tsitouras
Article
  • 13 Downloads

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).

Keywords

Initial value problem P-stability differential evolution semi-discretization of PDEs 

Mathematics Subject Classification

Primary 65L04 65L06 Secondary 65M20 90C59 

Notes

References

  1. 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)MathSciNetCrossRefGoogle Scholar
  2. 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)CrossRefGoogle Scholar
  3. 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)MathSciNetCrossRefGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 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)MathSciNetCrossRefGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 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)MathSciNetCrossRefGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 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)MathSciNetCrossRefGoogle Scholar
  10. 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)MathSciNetzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefGoogle Scholar
  12. 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) MathSciNetCrossRefGoogle Scholar
  13. 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) CrossRefGoogle Scholar
  14. 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)MathSciNetCrossRefGoogle Scholar
  15. 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)MathSciNetCrossRefGoogle Scholar
  16. 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)MathSciNetCrossRefGoogle Scholar
  17. 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)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Monovasilis, Th, Kalogiratou, Z., Simos, T.E.: Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inf. Sci. 7, 81–85 (2013)MathSciNetCrossRefGoogle Scholar
  19. 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)MathSciNetCrossRefGoogle Scholar
  20. 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)MathSciNetCrossRefGoogle Scholar
  21. 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) Google Scholar
  22. 22.
    Butcher, J.C.: Implicit Runge–Kutta processes. Math. Comput. 18, 50–64 (1964)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Butcher, J.C.: On Runge–Kutta processes of high order. J. Aust. Math. Soc. 4, 179–194 (1964)MathSciNetCrossRefGoogle Scholar
  24. 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)MathSciNetCrossRefGoogle Scholar
  25. 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)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Papageorgiou, G., Famelis, I.T., Tsitouras, C.: A P-stable singly diagonally implicit Runge–Kutta–Nyström method. Numer. Algorithms 17, 345–353 (1998)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tsitouras, Ch.: Stage reduction on P-stable Numerov type methods of eighth order. J. Comput. Appl. Math. 191, 297–305 (2006)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)zbMATHGoogle Scholar
  29. 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)MathSciNetCrossRefGoogle Scholar
  30. 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)MathSciNetzbMATHGoogle Scholar
  31. 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)MathSciNetCrossRefGoogle Scholar
  32. 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)MathSciNetCrossRefGoogle Scholar
  33. 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)CrossRefGoogle Scholar
  34. 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)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Papakostas, S.N., Tsitouras, Ch.: High phase-lag order Runge–Kutta and Nyström pairs. SIAM J. Sci. Comput. 21, 747–763 (1999)MathSciNetCrossRefGoogle Scholar
  36. 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)MathSciNetzbMATHGoogle Scholar
  37. 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)MathSciNetCrossRefGoogle Scholar
  38. 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)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Simos, T.E.: Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation. Acta Appl. Math. 110, 1331–1352 (2010)MathSciNetCrossRefGoogle Scholar
  40. 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)MathSciNetCrossRefGoogle Scholar
  41. 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)MathSciNetzbMATHGoogle Scholar
  42. 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)MathSciNetCrossRefGoogle Scholar
  43. 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)MathSciNetzbMATHGoogle Scholar
  44. 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)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Tsitouras, Ch., Famelis, I.T.: Symbolic derivation of Runge-Kutta-Nyström order conditions. J. Math. Chem. 46, 896–912 (2009)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Wolfram Research, Inc., Mathematica, Version 11.1. Wolfram Research, Inc., Champaign (2017)Google Scholar
  47. 47.
    Famelis, I.T., Papakostas, S.N., Tsitouras, Ch.: Symbolic derivation of Runge-Kutta order conditions. J. Symb. Comput. 37, 311–327 (2004)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Tsitouras, Ch., Famelis, I.T., Simos, T.E.: On modified Runge–Kutta trees and methods. Comput. Math. Appl. 62, 2101–2111 (2011)MathSciNetCrossRefGoogle Scholar
  49. 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)MathSciNetCrossRefGoogle Scholar
  50. 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)MathSciNetCrossRefGoogle Scholar
  51. 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)MathSciNetCrossRefGoogle Scholar
  52. 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)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Alonso-Mallo, I., Cano, B., Moreta, M.J.: Stability of Runge–Kutta–Nyström methods. J. Comput. Appl. Maths. 189, 120–131 (2006)CrossRefGoogle Scholar
  54. 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)MathSciNetCrossRefGoogle Scholar
  55. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  56. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Tsitouras, Ch.: Optimized explicit Runge–Kutta pair of orders 9(8). Appl. Numer. Math. 38, 123–134 (2001)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Tsitouras, Ch., Simos, T.E.: On ninth order, explicit Numerov type methods with constant coefficients. Mediterr. J. Math. 15 (2018) (Article No: 46) Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of ArtNingbo PolytechnicNingboChina
  2. 2.Department of Computer Science and Information EngineeringChang Jung Christian UniversityTainanTaiwan
  3. 3.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia
  4. 4.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina
  5. 5.Modern Computational MethodsUral Federal UniversityYekaterinburgRussian Federation
  6. 6.Section of Mathematics, Department of Civil EngineeringDemocritus University of ThraceXanthiGreece
  7. 7.AthensGreece
  8. 8.General DepartmentNational and Kapodistrian University of AthensEuboeaGreece

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