March  2019, 9(1): 71-84. doi: 10.3934/naco.2019006

A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems

1. 

Department of Mathematics, Beijing Jiaotong University Haibin College, Cangzhou, China

2. 

School of Science, Beijing Jiaotong University, Beijing, China

* Corresponding author: chenbingzhen6026@163.com

This paper was presented in the First Symposium on Machine Intelligence and Data Analytics (MIDA)-2017, Beijing, China, December 15-18, 2017.

Received  January 2018 Revised  April 2018 Published  October 2018

In this paper, we derive an implicit symmetric, symplectic and exponentially fitted Runge-Kutta-Nyström (ISSEFRKN) method. The new integrator ISSEFRKN2 is of fourth order and integrates exactly differential systems whose solutions can be expressed as linear combinations of functions from the set $\{\exp(λ t), \exp(-λ t)|λ∈ \mathbb{C}\}$, or equivalently $\{\sin(ω t), \cos(ω t)|λ = iω, ~ω∈ \mathbb{R}\}$. We analysis the periodicity stability of the derived method ISSEFRKN2. Some the existing implicit RKN methods in the literature are used to compare with ISSEFRKN2 for several oscillatory problems. Numerical results show that the method ISSEFRKN2 possess a more accuracy among them.

Citation: Wenjuan Zhai, Bingzhen Chen. A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 71-84. doi: 10.3934/naco.2019006
References:
[1]

P. Albrecht, The extension of the theory of A-methods to RK methods, In: Numerical Treatment of Differential Equations, Proceedings of the 4th Seminar NUMDIFF-4 (ed. K. Strehmel), Tuebner-Texte Zur Mathematik, Tuebner, Leipzig, (1987), 8–18.

[2]

P. Albrecht, A new theoretical approach to Runge Kutta methods, SIAM J. Numerical Anal., 24 (1987), 391-406.  doi: 10.1137/0724030.

[3]

R. A. Al-KhasawnehF. Ismail and M. Suleiman, Embedded diagonally implicit Runge-Kutta-Nyström 4(3) pair for solving special second-order IVPs, Appl. Math. Comput., 190 (2007), 1803-1814.  doi: 10.1016/j.amc.2007.02.067.

[4]

M. P. Calvo and J. M. Sanz-Serna, High-order symplectic Runge-Kutta-Nyström methods, SIAM J. Sci. Comput., 14 (1993), 1237-1252.  doi: 10.1137/0914073.

[5]

J. P. Coleman and L. Gr. Ixaru, P-stability and exponential-fitting methods for $y'' = f(x, y)$, IMA J. Numer. Anal., 16 (1996), 179-199.  doi: 10.1093/imanum/16.2.179.

[6]

J. M. Franco, Exponentially fitted explicit Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 167 (2004), 1-19.  doi: 10.1016/j.cam.2003.09.042.

[7]

J. M. Franco, Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems, Comput. Phys. Commun., 177 (2007), 479-492.  doi: 10.1016/j.cpc.2007.05.003.

[8]

E. Hairer, C. Lubich and G. Wanner, Symmetric Integration and Reversibility. In Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin, 2002. doi: 10.1007/978-3-662-05018-7.

[9]

L. Gr. Ixaru and G. Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht, Netherlands, 2004. doi: 10.1007/978-1-4020-2100-8.

[10]

S. N. Jator, Implicit third derivative Runge-Kutta-Nyström method with trigonometric coefficients, Numer. Algorithms, 70 (2015), 1-18.  doi: 10.1007/s11075-014-9938-5.

[11]

Z. Kalogiratou, Diagonally implicit trigonometrically fitted symplectic Runge-Kutta methods, Appl. Math. Comput., 219 (2013), 7406-7412.  doi: 10.1016/j.amc.2012.12.089.

[12]

Z. KalogiratouT. Monovasilis and T. E. Simos, A sixth order symmetric and symplectic diagonally implicit Runge-Kutta method, International Conference of Computational Metho. American Institute of Physics, 1618 (2014), 833-838.  doi: 10.1063/1.4897862.

[13]

K. W. MooN. SenuF. Ismail and N. M. Arifin, A zero-dissipative phase-fitted fourth order diagonally implicit Runge-Kutta-Nyström method for solving oscillatory problems, Math. Probl. Eng., 2014 (2014), 1-8.  doi: 10.1155/2014/985120.

[14]

B. Paternoster, Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials, Appl. Numer. Math., 28 (1998), 401-412.  doi: 10.1016/S0168-9274(98)00056-7.

[15]

M. Z. Qin and W. J. Zhu, Canonical Runge-Kutta-Nyström methods for second order ordinary differential equations, Comput. Math. Applic., 22 (1991), 85-95.  doi: 10.1016/0898-1221(91)90209-M.

[16]

J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview, Acta Numer., 1 (1992), 243-286.  doi: 10.1017/S0962492900002282.

[17]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994.

[18]

N. SenuM. SuleimanF. Ismail and M. Othman, A new diagonally implicit Runge-Kutta-Nyström method for periodic IVPs, WSEAS Trans. Math., 9 (2010), 679-688. 

[19]

P. W. SharpJ. M. Fine and K. Burrage, Two stage and three stage diagonally implicit Runge-Nutta-Nyström methods of orders three and four, IMA J. Numer. Anal., 10 (1990), 489-504.  doi: 10.1093/imanum/10.4.489.

[20]

T. E. Simos, An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions, Comput. Phys. Commun., 115 (1998), 1-8.  doi: 10.1016/S0010-4655(98)00088-5.

[21]

T. E. Simos and J. Vigo-Aguiar, Exponentially fitted symplectic integrator, Phys. Rev. E., 67 (2003), 1-7.  doi: 10.1103/PhysRevE.67.016701.

[22]

G. Vanden BergheH. De MeyerM. Van Daele and T. Van Hecke, Exponentially fitted Runge-Kutta methods, J. Comput. Appl. Math., 125 (2000), 107-115.  doi: 10.1016/S0377-0427(00)00462-3.

[23]

G. Vanden BergheM. Van Daele and H. Van de Vyver, Exponential fitted Runge-Kutta methods of collocation type: fixed or variable knot points?, J. Comput. Appl. Math., 159 (2003), 217-239.  doi: 10.1016/S0377-0427(03)00450-3.

[24]

P. J. Van der Houwen and B. P. Sommeijer, Diagonally implicit Runge-Nutta-Nyström methods for oscillating problems, SIAM J. Numer. Anal., 26 (1989), 414-429.  doi: 10.1137/0726023.

[25]

X. You and B. Chen, Symmetric and symplectic exponentially fitted Runge-Kutta(-Nyström) methods for Hamiltonian problems, Math. Comput. Simul., 94 (2013), 76-95.  doi: 10.1016/j.matcom.2013.05.010.

show all references

References:
[1]

P. Albrecht, The extension of the theory of A-methods to RK methods, In: Numerical Treatment of Differential Equations, Proceedings of the 4th Seminar NUMDIFF-4 (ed. K. Strehmel), Tuebner-Texte Zur Mathematik, Tuebner, Leipzig, (1987), 8–18.

[2]

P. Albrecht, A new theoretical approach to Runge Kutta methods, SIAM J. Numerical Anal., 24 (1987), 391-406.  doi: 10.1137/0724030.

[3]

R. A. Al-KhasawnehF. Ismail and M. Suleiman, Embedded diagonally implicit Runge-Kutta-Nyström 4(3) pair for solving special second-order IVPs, Appl. Math. Comput., 190 (2007), 1803-1814.  doi: 10.1016/j.amc.2007.02.067.

[4]

M. P. Calvo and J. M. Sanz-Serna, High-order symplectic Runge-Kutta-Nyström methods, SIAM J. Sci. Comput., 14 (1993), 1237-1252.  doi: 10.1137/0914073.

[5]

J. P. Coleman and L. Gr. Ixaru, P-stability and exponential-fitting methods for $y'' = f(x, y)$, IMA J. Numer. Anal., 16 (1996), 179-199.  doi: 10.1093/imanum/16.2.179.

[6]

J. M. Franco, Exponentially fitted explicit Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 167 (2004), 1-19.  doi: 10.1016/j.cam.2003.09.042.

[7]

J. M. Franco, Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems, Comput. Phys. Commun., 177 (2007), 479-492.  doi: 10.1016/j.cpc.2007.05.003.

[8]

E. Hairer, C. Lubich and G. Wanner, Symmetric Integration and Reversibility. In Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin, 2002. doi: 10.1007/978-3-662-05018-7.

[9]

L. Gr. Ixaru and G. Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht, Netherlands, 2004. doi: 10.1007/978-1-4020-2100-8.

[10]

S. N. Jator, Implicit third derivative Runge-Kutta-Nyström method with trigonometric coefficients, Numer. Algorithms, 70 (2015), 1-18.  doi: 10.1007/s11075-014-9938-5.

[11]

Z. Kalogiratou, Diagonally implicit trigonometrically fitted symplectic Runge-Kutta methods, Appl. Math. Comput., 219 (2013), 7406-7412.  doi: 10.1016/j.amc.2012.12.089.

[12]

Z. KalogiratouT. Monovasilis and T. E. Simos, A sixth order symmetric and symplectic diagonally implicit Runge-Kutta method, International Conference of Computational Metho. American Institute of Physics, 1618 (2014), 833-838.  doi: 10.1063/1.4897862.

[13]

K. W. MooN. SenuF. Ismail and N. M. Arifin, A zero-dissipative phase-fitted fourth order diagonally implicit Runge-Kutta-Nyström method for solving oscillatory problems, Math. Probl. Eng., 2014 (2014), 1-8.  doi: 10.1155/2014/985120.

[14]

B. Paternoster, Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials, Appl. Numer. Math., 28 (1998), 401-412.  doi: 10.1016/S0168-9274(98)00056-7.

[15]

M. Z. Qin and W. J. Zhu, Canonical Runge-Kutta-Nyström methods for second order ordinary differential equations, Comput. Math. Applic., 22 (1991), 85-95.  doi: 10.1016/0898-1221(91)90209-M.

[16]

J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview, Acta Numer., 1 (1992), 243-286.  doi: 10.1017/S0962492900002282.

[17]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994.

[18]

N. SenuM. SuleimanF. Ismail and M. Othman, A new diagonally implicit Runge-Kutta-Nyström method for periodic IVPs, WSEAS Trans. Math., 9 (2010), 679-688. 

[19]

P. W. SharpJ. M. Fine and K. Burrage, Two stage and three stage diagonally implicit Runge-Nutta-Nyström methods of orders three and four, IMA J. Numer. Anal., 10 (1990), 489-504.  doi: 10.1093/imanum/10.4.489.

[20]

T. E. Simos, An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions, Comput. Phys. Commun., 115 (1998), 1-8.  doi: 10.1016/S0010-4655(98)00088-5.

[21]

T. E. Simos and J. Vigo-Aguiar, Exponentially fitted symplectic integrator, Phys. Rev. E., 67 (2003), 1-7.  doi: 10.1103/PhysRevE.67.016701.

[22]

G. Vanden BergheH. De MeyerM. Van Daele and T. Van Hecke, Exponentially fitted Runge-Kutta methods, J. Comput. Appl. Math., 125 (2000), 107-115.  doi: 10.1016/S0377-0427(00)00462-3.

[23]

G. Vanden BergheM. Van Daele and H. Van de Vyver, Exponential fitted Runge-Kutta methods of collocation type: fixed or variable knot points?, J. Comput. Appl. Math., 159 (2003), 217-239.  doi: 10.1016/S0377-0427(03)00450-3.

[24]

P. J. Van der Houwen and B. P. Sommeijer, Diagonally implicit Runge-Nutta-Nyström methods for oscillating problems, SIAM J. Numer. Anal., 26 (1989), 414-429.  doi: 10.1137/0726023.

[25]

X. You and B. Chen, Symmetric and symplectic exponentially fitted Runge-Kutta(-Nyström) methods for Hamiltonian problems, Math. Comput. Simul., 94 (2013), 76-95.  doi: 10.1016/j.matcom.2013.05.010.

Figure 1.  Periodicity regions for the method ISSEFRKN2.
Figure 2.  Maximum global error in the solution for Problem 1.
Figure 3.  Maximum global error in the solution for Problem 2.
Figure 4.  Maximum global error in the solution for problem 3 with $\varepsilon = 0$.
Figure 5.  Maximum global error in the solution for problem 3 with $\varepsilon = 10^{-3}$.
Figure 6.  Maximum global error in the solution for problem 4.
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