Frequency-dependent Symmetric Hybrid Multistep Method with Bounded Amplitude Error for Stiff and Oscillatory ODEs
Abstract
This paper considers a new fourth-order frequency-dependent symmetric hybrid linear multistep method for oscillatory second-order differential equations. The proposed method achieves minimal phase-lag and bounded amplitude error, which accurately reproduces orbital trajectories, and requires fewer function evaluations. Numerical experiment confirms improved efficiency, accuracy, and long-term stability over some existing methods in the literature.
Downloads
References
Lambert, J. D., & Watson, I. A. (1976). Symmetric multistep methods for periodic initial value problems. IMA Journal of Applied Mathematics, 18(2), 189–202. https://doi.org/10.1093/imamat/18.2.189
Wang, Z., Zhao, D., Dai, Y., & Wu, D. (2005). An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial-value problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2058), 1639–1658. https://doi.org/10.1098/rspa.2004.1438
Fatunla, S. O., Ikhile, M. N. O., & Otunta, F. O. (1999). A class of P-stable linear multistep numerical methods. International Journal of Computer Mathematics, 72(1), 1–13. https://doi.org/10.1080/00207169908804830
Hairer, E. (1979). Unconditionally stable methods for second order differential equations. Numerische Mathematik, 32(4), 373–379. https://doi.org/10.1007/BF01401041
Cash, J. R. (1981). High order P-stable formulae for the numerical integration of periodic initial value problems. Numerische Mathematik, 37(3), 355–370. https://doi.org/10.1007/BF01400315
Shokri, A. (2014). One and two-step new hybrid methods for the numerical solution of first order initial value problems. Acta Universitatis Matthiae Belii, Series Mathematics Online Edition, 45–58.
Felix, I. C., & Okuonghae, R. I. (2019). On the generalisation of Padé approximation approach for the construction of P-stable hybrid linear multistep methods. International Journal of Applied and Computational Mathematics, 5(3), 93. https://doi.org/10.1007/s40819-019-0685-0
Felix, I. C., & Okuonghae, R. I. (2018). On the construction of P-stable hybrid multistep methods for second order ODEs. Far East Journal of Applied Mathematics, 99(3), 259–273. https://doi.org/10.17654/AM099030259
Brusa, L., & Nigro, L. (1980). A one-step method for direct integration of structural dynamic equations. International Journal for Numerical Methods in Engineering, 15(5), 685–699. https://doi.org/10.1002/nme.1620150506
Simos, T. E. (1999). Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation. Computers & Chemistry, 23(5), 439–446. https://doi.org/10.1016/S0097-8485(99)00028-5
Alolyan, I., & Simos, T. E. (2011). A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Computers & Mathematics with Applications, 62(10), 3756–3774. https://doi.org/10.1016/j.camwa.2011.09.025
Shokri, A., Mehdizadeh Khalsaraei, M., & Atashyar, A. (2020). A new two-step hybrid singularly P-stable method for the numerical solution of second-order IVPs with oscillating solutions. Iranian Journal of Mathematical Chemistry, 11(2), 113–132. https://doi.org/10.1002/cmm4.1116
Ibrahimov, V. R., Mehdiyeva, G. Y., Yue, X.-G., Kaabar, M. K. A., Noeiaghdam, S., & Juraev, D. A. (2021). Novel symmetric numerical methods for solving symmetric mathematical problems. International Journal of Circuits, Systems and Signal Processing, 15(1), 1545–1557. https://doi.org/10.46300/9106.2021.15.167
Dahlquist, G. (1978). On accuracy and unconditional stability of linear multistep methods for second order differential equations. BIT Numerical Mathematics, 18(2), 133–136. https://doi.org/10.1007/BF01931689
Neta, B. (2005). P-stable symmetric super-implicit methods for periodic initial value problems. Computers & Mathematics with Applications, 50(5–6), 701–705. https://doi.org/10.1016/j.camwa.2005.04.013
Anake, T. A., Felix, I. C., & Ogundile, O. P. (2018). Higher order super-implicit hybrid multistep methods for second order differential equations. International Journal of Mechanical Engineering and Technology, 9(8), 1384–1392.
Sun, B., Lin, C.-L., & Simos, T. E. (2023). Solution to quantum chemistry problems using a phase-fitting, singularly P-stable, cost-effective two-step approach with disappearing phase-lag derivatives up to order 5. Journal of Mathematical Chemistry, 61(3), 455–489. https://doi.org/10.1007/s10910-022-01416-w
Stiefel, E., & Bettis, D. G. (1969). Stabilization of Cowell’s method. Numerische Mathematik, 13(2), 154–175. https://doi.org/10.1007/BF02163234
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
