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Construction of an optimized explicit Runge-Kutta-Nyström method for the numerical solution of oscillatory initial value problems. (English) Zbl 1222.65066
Summary: An explicit optimized Runge-Kutta-Nyström method with four stages and fifth algebraic order is developed. The produced method has variable coefficients with zero phase-lag, zero amplification factor and zero first derivative of the amplification factor. We provide an analysis of the local truncation error of the new method. We also measure the efficiency of the new method in comparison to other numerical methods through the integration of the two-body problem with various eccentricities and three other periodical/oscillatory initial value problems.

65L06Multistep, Runge-Kutta, and extrapolation methods
Full Text: DOI
[1] Aceto, L.; Sestini, A.: Numerical aspects of the coefficient computation for lmms, Journal of numerical analysis, industrial and applied mathematics 3, 181-191 (2008) · Zbl 1255.65134
[2] Chatterjee, Samrat; Isaiay, Marco; Bonay, Francesca; Badinoy, Guido; Venturino, Ezio: Modelling environmental influences on wanderer spiders in the langhe region (Piemonte - NW Italy), Journal of numerical analysis, industrial and applied mathematics 3, 193-209 (2008) · Zbl 1166.92037
[3] Dagnino, C.; Demichelis, V.; Lamberti, P.: A nodal spline collocation method for the solution of Cauchy singular integral equations, Journal of numerical analysis, industrial and applied mathematics, 211-220 (2008) · Zbl 1176.65153
[4] Enachescu, C.: Approximation capabilities of neural networks, Journal of numerical analysis, industrial and applied mathematics, 221-230 (2008) · Zbl 1166.92301
[5] Frederich, O.; Wassen, E.; Thiele, F.: Prediction of the flow around a short wall-mounted finite cylinder using LES and DES, Journal of numerical analysis, industrial and applied mathematics, 231-247 (2008) · Zbl 1166.76024
[6] Ogata, H.: Fundamental solution method for periodic plane elasticity, Journal of numerical analysis, industrial and applied mathematics, 249-267 (2008) · Zbl 1166.74005
[7] An, Phan Thanh: Some computational aspects of Helly-type theorems, Journal of numerical analysis, industrial and applied mathematics, 269-274 (2008) · Zbl 1168.52006
[8] Verhoeven, A.; Tasic, B.; Beelen, T. G. J.; Maten, E. J. W. Ter; Mattheij, R. M. M.: BDF compound-fast multirate transient analysis with adaptive stepsize control, Journal of numerical analysis, industrial and applied mathematics 3, 275-297 (2008) · Zbl 1170.65069
[9] Van De Vyver, Hans: A $5(3)$ pair of explicit Runge--Kutta--Nyström methods for oscillatory problems, Mathematical and computer modelling 45, 708-716 (2006) · Zbl 1165.65368 · doi:10.1016/j.mcm.2006.07.016
[10] Simos, T. E.; Vigo-Aguiar, J.: Exponentially fitted symplectic integrator, Physical review E 67 (2003) · Zbl 1196.65123
[11] Tocino, A.; Vigo-Aguiar, J.: Symplectic conditions for exponential Fitting Runge--Kutta--Nyström methods, Mathematical and computer modelling 42, 873-876 (2005) · Zbl 1086.65120 · doi:10.1016/j.mcm.2005.09.015
[12] Ernst, Hairer; Wanner, Gerhard; Nørsett, Syvert P.: Solving ordinary differential equations, I, nonstiff problems, Springer series in computational mathematics 298 (2008) · Zbl 0789.65048
[13] Aguiar, J. Vigo; Simos, T. E.: An exponentially fitted and trigonometrically fitted method for the numerical solution of orbital problems, The astronomical journal 122, No. 3, 1656-1660 (2001)
[14] Simos, T. E.: Chemical modelling - applications and theory, vol. 1, Specialist periodical reports 1, 32-140 (2000)
[15] Simos, T. E.; Tsitouras, Ch.: A P-stable eighth order method for the numerical integration of periodic initial value problems, Journal of computational physics 130, 123-128 (1997) · Zbl 0870.65072 · doi:10.1006/jcph.1996.5567
[16] Dormand, J. R.; El-Mikkawy, M. E. A.; Prince, P. J.: Families of Runge--Kutta- Nyström formulae, IMA journal of numerical analysis 7, 235-250 (1987) · Zbl 0624.65059 · doi:10.1093/imanum/7.2.235
[17] Papadopoulos, D. F.; Anastassi, Z. A.; Simos, T. E.: A phase-fitted Runge--Kutta--Nyström method for the numerical solution of initial value problems with oscillating solutions, Computer physics communications 180, 1839-1846 (2009) · Zbl 1197.65086 · doi:10.1016/j.cpc.2009.05.014
[18] Engeln-Mullges, Gisela; Uhlig, Frank: Numerical algorithms with Fortran, (1996) · Zbl 0857.65002
[19] Gr, L.; Ixaru; Rizea, M.: A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies, Computer physics communications 19, 23-27 (1980)
[20] Cooley, J. W.: An improved eigenvalue corrector formula for solving the Schrödinger equation for central fields, Mathematics of computation 15, 63 (1961) · Zbl 0122.35902 · doi:10.2307/2003025
[21] Blatt, J. M.: Practical points concerning the solution of the Schrödinger equation, Journal of computational physics 1, 382 (1967) · Zbl 0182.49702 · doi:10.1016/0021-9991(67)90046-0