SIAM Journal on Numerical Analysis
Analysis of four numerical schemes for a nonlinear Klein-Gordon equation
Applied Mathematics and Computation
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Canonical Runge-Kutta-Nystro¨m methods of orders five and six
Journal of Computational and Applied Mathematics
Long-Time-Step Methods for Oscillatory Differential Equations
SIAM Journal on Scientific Computing
Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations
SIAM Journal on Numerical Analysis
Exponentially fitted explicit Runge-Kutta-Nyström methods
Journal of Computational and Applied Mathematics
New methods for oscillatory systems based on ARKN methods
Applied Numerical Mathematics
Scheifele two-step methods for perturbed oscillators
Journal of Computational and Applied Mathematics
Structure-Preserving Algorithms for Oscillatory Differential Equations
Structure-Preserving Algorithms for Oscillatory Differential Equations
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The numerical integration of Hamiltonian systems with multi-frequency and multidimensional oscillatory solutions is encountered frequently in many fields of the applied sciences. In this paper, we firstly summarize the extended Runge---Kutta---Nyström (ERKN) methods proposed by Wu et al. (Comput. Phys. Comm. 181:1873---1887, (2010)) for multi-frequency and multidimensional oscillatory systems and restate the order conditions and symplecticity conditions for the explicit ERKN methods. Secondly, we devote to exploring the explicit symplectic multi-frequency and multidimensional ERKN methods of order five based on the symplecticity conditions and order conditions. A five-stage explicit symplectic multi-frequency and multidimensional ERKN method of order five with some small residuals is proposed and its stability and phase properties are analyzed. It is shown that the new method is dispersive of order six. Numerical experiments are carried out and the numerical results demonstrate that the new method is much more efficient than the methods appeared in the scientific literature.