Extended RKN-type methods for numerical integration of perturbed oscillators

An efficient high-order explicit scheme for solving Hamiltonian nonlinear wave equations

“…This leads to the ERKN integrators for (23) as follows [25]. Usually, this method (27) can be compactly displayed by the Butcher tableau of coefficients It is very important to note that when M ¼ 0 dÂd , the scheme (27) reduces to a classical RKN method.…”

“…Taking account of the oscillatory structure generated by the linear part My, Wu et al [24] proposed a standard form of the multi-frequency and multidimensional ERKN methods for (23) and derived the corresponding order conditions based on the extended Nyström tree theory, a tri-colored rooted tree theory (readers are referred to [24,25]). A remarkable advantage of ERKN methods is that both their internal stages and updates take into account the special oscillatory structures of the system (23) introduced by the linear term My, therefore, they can naturally integrate the multi-frequency and multidimensional oscillatory homogeneous system y 00 ðtÞ þ MyðtÞ ¼ 0 exactly.…”

ERKN integrators for systems of oscillatory second-order differential equations

“…But both the order conditions and the analysis for stability and phase properties of the methods in his paper [10] are based on the one-dimensional theory. And the approach in Yang et al [15] on the ERKN methods where the matrix M is diagonal with nonnegative entries, is likewise. This is not satisfactory and therefore, we are motivated to develop multidimensional ERKN methods for systems of perturbed oscillators (8) in an effective way, where the matrix M is not required to be symmetric and positive semi-definite.…”

“…This leads to the following definition of ERKN methods proposed by H. Yang et al [15]. Definition 2.1.…”

“…are satisfied, where τ is an extended Nyström tree defined in [15] corresponding to an elementary differential F (y) of the function f (y) at y n , the φ-functions φ j (v) in scalar variable v are defined in [7], the functions ρ(τ ), γ (τ ) and Φ(τ ) are defined in [15].…”