Extended Runge–Kutta–Nystrom Methods Research Articles

Symmetric collocation ERKN methods for general second-order oscillators

For the numerical solution of the general second-order oscillatory system $$y''+ M y = f(y,y')$$, You et al. (Numer Algorithm 66:147–176, 2014) proposed the extended Runge–Kutta–Nystrom (ERKN) methods. This paper is devoted to symmetric collocation ERKN methods of Gauss and Lobatto IIIA types by Lagrange interpolation. Linear stability of the new ERKN methods is analyzed. Numerical experiments show the high effectiveness of the new ERKN methods compared to their RKN counterparts.

Diagonal implicit symplectic extended RKN methods for solving oscillatory Hamiltonian systems

This paper studies diagonal implicit symplectic extended Runge–Kutta–Nystrom (ERKN) methods for solving the oscillatory Hamiltonian system $$H(q,p)=\dfrac{1}{2}p^\mathrm{T}p+\dfrac{1}{2}q^\mathrm{T}Mq+U(q)$$ . Based on symplecticity conditions and order conditions, we construct some diagonal implicit symplectic ERKN methods. The stability of the obtained methods is discussed. Three numerical experiments are carried out to show the performance of the methods. It follows from the numerical results that the new diagonal implicit symplectic methods are more effective than RKN methods when applied to the oscillatory Hamiltonian system.

Order conditions for RKN methods solving general second-order oscillatory systems

This paper proposes and investigates the multidimensional extended Runge-Kutta-Nystrom (ERKN) methods for the general second-order oscillatory system y? + My = f(y, y?) where M is a positive semi-definite matrix containing implicitly the frequencies of the problem. The work forms a natural generalization of our previous work on ERKN methods for the special system y? + My = f(y) (H. Yang et al. Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777---1794 and X. Wu et al., ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Comm. 181 (2010) 1873---1887). The new ERKN methods, with coefficients depending on the frequency matrix M, incorporate the special structure of the equation brought by the term My into both internal stages and updates. In order to derive the order conditions for the ERKN methods, an extended Nystrom tree (EN-tree) theory is established. The results of numerical experiments show that the new ERKN methods are more efficient than the general-purpose RK methods and the adapted RKN methods with the same algebraic order in the literature.