A multi-point numerical integrator with trigonometric coefficients for initial value problems with periodic solutions

J. O. Ehigie ; S. N. Jator ; S. A. Okunuga

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 3, pp. 329-344.

Résumé

Based on a collocation technique, we introduce a unifying approach for deriving a family of multi-point numerical integrators with trigonometric coefficients for the numerical solution of periodic initial value problems. A practical $3$-point numerical integrator is presented, whose coefficients are generalizations of classical linear multistep methods such that the coefficients are functions of an estimate of the angular frequency $\omega$. The collocation technique yields a continuous method, from which the main and complementary methods are recovered and expressed as a block matrix finite difference formula which integrates a second order differential equation over non-overlapping intervals without predictors. Some properties of the numerical integrator are investigated and presented. Numerical examples are given to illustrate the accuracy of the method.

Bibliographie

[1] Aceto L., Ghelardoni P., Magherini C., “PGSCM: a family of P-stable boundary value methods for second-order initial value problems”, J. of Comput. and Appl. Math., 236:16 (2012), 3857–3868 | DOI | MR | Zbl

[2] Brugnano L., Trigiante D., Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach, Amsterdam, 1998 | MR

[3] Chawla M. M., Canonical Runge–Kutta–Nystrom methods, Thesis Ph.D. de Vallidolid, Universidad de Vallidolid, 1992

[4] Chawla M. M., Rao P. S., “A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: explicit methods”, J. of Comput. and Appl. Math., 15 (1986), 329–337 | DOI | MR | Zbl

[5] Chawla M. M., Rao P. S., Neta B., “Two-step fourth-order P-stable methods with phase lag of order six for $y'=f(t,y)$”, J. of Comput. and Appl. Math., 6 (1986), 233–236 | DOI | MR

[6] Coleman J. P., “Numerical methods for $y''=f(x,y)$ via rational approximations for the cosine”, IMA J. of Numerical Analysis, 9 (1989), 145–165 | DOI | MR | Zbl

[7] Coleman J. P., Ixaru L. G., “P-stability and exponential fitting methods for $y''=f(x,y)$”, IMA J. of Numerical Analysis, 16 (1996), 179–199 | DOI | MR | Zbl

[8] Ehigie J. O., Okunuga S. A., Sofoluwe A. B., “3-point block methods for direct integration of second order ordinary differential equations”, Advances in Numerical Analysis, 2011 (2011), Article ID 513148 | DOI | MR | Zbl

[9] Ehigie J. O., Jator S. N., Sofoluwe A. B., Okunuga S. A., “Boundary value technique for initial value problems with continuous second derivative multistep method of Enright”, Comput. and Appl. Math., 33 (2014), 81–93 | DOI | MR | Zbl

[10] Ehigie J. O., Okunuga S. A., “A stiffly stable second derivative block multistep formula with Chebyshev collocation points for stiff problems”, Int. J. of Pure and Appl. Math., 96:4 (2014), 457–481 | DOI | MR

[11] Fang Y., Song Y., Wu X., “A robust trigonometrically fitted embedded pair for pertubed oscillators”, J. of Comp. and Appl. Math., 225:2 (2009), 347–355 | DOI | MR | Zbl

[12] Fatunla S. O., “Block methods for second order IVPs”, Int. J. of Comp. Math., 41 (1991), 55–63 | DOI | Zbl

[13] Franco J. M., “An explicit hybrid method of Numerov-type for second-order periodic initial value problems”, J. of Comp. and Appl. Math., 59 (1995), 79–90 | DOI | MR | Zbl

[14] Franco J. M., “Runge–Kutta–Nystrom methods adapted to the numerical integration of perturbed oscillators”, Comp. Physics Communications, 147:3 (2002), 770–787 | DOI | MR | Zbl

[15] Gautschi W., “Numerical integration of ordinary differential equations based on trigonometric polynomials”, Numerische Mathematik, 1961, no. 3, 381–397 | DOI | MR | Zbl

[16] Hairer E., Wanner G., Solving Ordinary Differential Equations, v. II, Stiff and Differential Algebraic Problems, Second Revised Edition, Springer Verlag, Germany, 1996 | MR | Zbl

[17] Hindmarsh A. C., “Odepack, a systematized collection of ODE solvers”, Scientific Computing, v. 1, eds. R. S. Stepleman et al., North-Holland, Amsterdam, 1983, 55–64 | MR

[18] Ixaru L. Gr., Vanden Berghe G., Exponential Fitting, Kluwer Academic Publishers, 2004 | MR | Zbl

[19] Ixaru L. Gr., Vanden Berghe G., De Meyer H., “Frequency evaluation in exponential fitting multistep algorithms for ODEs”, J. of Comput. and Appl. Math., 140:1–2 (2002), 423–434 | DOI | MR | Zbl

[20] Jator S. N., “Solving second order initial value problems by a hybrid multistep methods without predictors”, Appl. Math. and Comput., 217 (2010), 4036–4046 | MR | Zbl

[21] Jator S. N., Swindle S., French R., “Trigonometrically fitted block Numerov type method for $y''=f(x,y,y')$”, Numer. Algor., 62:1 (2013), 13–26 | DOI | MR | Zbl

[22] Lambert J. D., Computational Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1973 | Zbl

[23] Li J., Wang X., “Multistep hybrid methods for special second-order differential equations $y''=f(t,y(t))$”, Numer. Algor., 73:3 (2016), 711–733 | DOI | MR | Zbl

[24] Lyche T., “Chebyshevian multistep methods for ordinary differential equations”, Numerische Mathematik, 19 (1972), 65–75 | DOI | MR | Zbl

[25] Nguyen H. S., Sidje R. B., Cong N. H., “Analysis of trigonometrically fitted implicit Runge–Kutta methods”, J. of Comput. and Appl. Math., 198:1 (2007), 187–207 | DOI | MR | Zbl

[26] Ngwane F. F., Jator S. N., “Solving oscillatory problems using a block hybrid trigonometrically fitted method with two off-step point”, Proceedings of the Ninth MSU-UAB Conference on Differential Equations and Computational Simulations, Electronic J. of Differential Equations, 20, 2013, 119–132 | MR

[27] Paternoster B., “Runge–Kutta (Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials”, Appl. Numer. Math., 28 (1998), 401–412 | DOI | MR | Zbl

[28] Paternoster B., “Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday”, Comput. Physics Communications, 183:12 (2012), 2499–2512 | DOI | MR | Zbl

[29] Ramos H., Kalogiratou Z., Monovalis Th., Simos T. E., “An optimized two-step hybrid block method for solving general second order initial value problem”, Numer. Algor., 72:4 (2016), 1089–1102 | DOI | Zbl

[30] Shampine L. F., Watts H. A., “Block implicit one-step methods”, Math. of Comput., 23 (1969), 731–740 | DOI | Zbl

[31] Simos T. E., “An exponentially-fitted Runge–Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions”, Comput. Physics Communications, 115:1 (1998), 1–8 | DOI | MR | Zbl

[32] Simos T. E., Dimas E., Sideridis A. B., “A Runge–kutta method for the numerical integration of special second-order periodic initial-value problems”, J. of Comput. and Appl. Math., 51:3 (1994), 317–326 | DOI | MR | Zbl

[33] Vigo-Aguiar J., Ramos H., “On the choice of the frequency in trigonometrically-fitted methods for periodic problems”, J. of Comput. and Appl. Math., 277:15 (2015), 94–105 | DOI | MR | Zbl

[34] Wang Z., Zhao D., Dai Y., Wu D., “An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial-value problems”, Proc. of the Royal Society A, 461 (2005), 1639–1658 | DOI | MR | Zbl