Multidimensional adapted Runge-Kutta-Nyström methods for oscillatory systems

Abstract For the general multidimensional perturbed oscillators y ″ + K y = f ( y , y ′ ) with K ∈ R d × d , the order conditions for the ARKN methods are presented by X. Wu et al. [Order conditions for ARKN methods solving oscillatory systems, Comput. Phys. Comm. 180 (2009) 2250–2257]. These methods integrate exactly the multidimensional unperturbed oscillators and are highly efficient when the perturbing function is small. In this paper, we are concerned with the analysis of the concrete multidimensional ARKN methods based on the order conditions for the multidimensional ARKN methods. We extent the matrix K to a general case, in which K is not required to be symmetric. Numerical experiments demonstrate that the novel multidimensional ARKN methods presented in this paper are more efficient compared with some well-know RKN methods in the scientific literature.

[1]  Bin Wang,et al.  ERKN integrators for systems of oscillatory second-order differential equations , 2010, Comput. Phys. Commun..

[2]  Ben P. Sommeijer,et al.  Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions , 1987 .

[3]  Hans Van de Vyver,et al.  Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems , 2005, Comput. Phys. Commun..

[4]  Xinyuan Wu,et al.  Note on derivation of order conditions for ARKN methods for perturbed oscillators , 2009, Comput. Phys. Commun..

[5]  J. M. Franco.  Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators , 2002 .

[6]  Jesús Vigo-Aguiar,et al.  An exponentially-fitted high order method for long-term integration of periodic initial-value problems , 2001 .

[7]  J. M. Franco.  New methods for oscillatory systems based on ARKN methods , 2006 .

[8]  Marlis Hochbruck,et al.  A Gautschi-type method for oscillatory second-order differential equations , 1999, Numerische Mathematik.

[9]  Ana B. González,et al.  New methods for oscillatory problems based on classical codes , 2002 .

[10]  Xinyuan Wu,et al.  A new pair of explicit ARKN methods for the numerical integration of general perturbed oscillators , 2007 .

[11]  Jianlin Xia,et al.  Order conditions for ARKN methods solving oscillatory systems , 2009, Comput. Phys. Commun..

[12]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[13]  Xinyuan Wu,et al.  Trigonometrically-fitted ARKN methods for perturbed oscillators , 2008 .

[14]  J. M. Franco.  A 5(3) pair of explicit ARKN methods for the numerical integration of perturbed oscillators , 2003 .