Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations
SIAM Journal on Numerical Analysis
Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems
SIAM Journal on Numerical Analysis
New methods for oscillatory systems based on ARKN methods
Applied Numerical Mathematics
Spectral Semi-discretisations of Weakly Non-linear Wave Equations over Long Times
Foundations of Computational 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
| Hi-index | 0.00 |
In this paper, we are concerned with the error analysis for the two-step extended Runge-Kutta-Nyström-type (TSERKN) methods [Comput. Phys. Comm. 182 (2011) 2486---2507] for multi-frequency and multidimensional oscillatory systems y驴(t) + My(t) = f(t, y(t)), where high-frequency oscillations in the solutions are generated by the linear part My(t). TSERKN methods extend the two-step hybrid methods [IMA J. Numer. Anal. 23 (2003) 197---220] by reforming both the internal stages and the updates so that they are adapted to the oscillatory properties of the exact solutions. However, the global error analysis for the TSERKN methods has not been investigated. In this paper we construct a new three-stage explicit TSERKN method of order four and present the global error bound for the new method, which is proved to be independent of 驴M驴 under suitable assumptions. This property of our new method is very important for solving highly oscillatory systems (1), where 驴M驴 may be arbitrarily large. We also analyze the stability and phase properties for the new method. Numerical experiments are included and the numerical results show that the new method is very competitive and promising compared with the well-known high quality methods proposed in the scientific literature.