Hermite-Birkhoff-Obrechkoff four-stage four-step ODE solver of order 14 with quantized step size. (English) Zbl 1153.65071

Summary: A four-stage Hermite-Birkhoff-Obrechkoff method of order 14 with four quantized variable steps, denoted by HBOQ(14)4, is constructed for solving non-stiff systems of first-order differential equations of the form \(y{^{\prime}}=f(t,y)\) with initial conditions \(y(t_{0})=y_{0}\). Its formula uses \(y{^{\prime}}, y{^{\prime\prime}}\) and \(y^{\prime \prime \prime }\) as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge-Kutta-type order conditions which are reorganized into linear Vandermonde-type systems.
To reduce overhead, simple formulae are derived only once to obtain the values of Hermite-Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized step size ratios. The step size is controlled by a local error estimator. When programmed in C\(++\), HBOQ(14)4 is superior to the Dormand-Prince Runge-Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ordinary differential equation solvers at stringent tolerances. When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of the number of steps, CPU time, and maximum global error. The code is available on the URL www.site.uottawa.ca/\(^{\sim}\)remi.


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
68W30 Symbolic computation and algebraic computation
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations


ode113; Matlab; HBOQ(14)4
Full Text: DOI


[1] Barrio, R.; Blesa, F.; Lara, M., VSVO formulation of the Taylor method for the numerical solution of odes, Comput. math. appl., 50, 93-111, (2005) · Zbl 1085.65056
[2] Binney, J.; Tremaine, S., Galactic dynamics, (1987), Princeton Univ. Press · Zbl 1130.85301
[3] Broucke, R.A., Numerical integration of periodic orbits in the main problem of artificial satellite theory, Celestial mech. dynam. astronom., 58, 99-123, (1994)
[4] P.J. Bryant, Nonlinear wave groups in deep water (manuscript) · Zbl 0432.76022
[5] Butcher, J.C., Coefficients for the study of runge – kutta integration processes, J. aust. math. soc., 3, 185-201, (1963) · Zbl 0223.65031
[6] Butcher, J.C., A modified multistep method for the numerical integration of ordinary differential equations, J. assoc. comput. Mach., 12, 124-135, (1965) · Zbl 0125.07102
[7] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations I. nonstiff problems, section III.8, (1993), Springer-Verlag Berlin · Zbl 0789.65048
[8] Hénon, H.; Heiles, C., The applicability of the third integral of motion. some numerical examples, Astron. J., 69, 73-79, (1964)
[9] Hull, T.E.; Enright, W.H.; Fellen, B.M.; Sedgwick, A.E., Comparing numerical methods for ordinary differential equations, SIAM J. numer. anal., 9, 603-637, (1972) · Zbl 0221.65115
[10] Lambert, J.D., Computational methods in ordinary differential equations, (1973), Wiley London, (Chapter 5) · Zbl 0258.65069
[11] T. Nguyen-Ba, R. Vaillancourt, Hermite-Birkhoff differential equation solvers, in: Scientific Proceedings of Riga Technical University, 5-th series: Computer Science, 46-th thematic issue, 21, 2004, pp. 47-64
[12] Nguyen-Ba, T.; Yagoub, H.; Zhang, Y.; Vaillancourt, R., Variable-step variable-order 3-stage hermite – birkhoff – obrechkoff ODE solver of order 4 to 14, Can. appl. math. quarterly, 14, 4, 413-437, (2006) · Zbl 1138.65058
[13] T. Nguyen-Ba, P.W. Sharp, H. Yagoub, R. Vaillancourt, Hermite-Birkhoff-Obrechkoff 3-stage 4-step ODE Solver of order 14 with quantized stepsize, Can. Appl. Math. Q. (in press) · Zbl 1220.65086
[14] Obrechkoff, N., Neue quadraturformeln, Abh. preuss. akad. wiss. math. nat. kl., 4, 1-20, (1940) · Zbl 0024.02602
[15] Prince, P.J.; Dormand, J.R., High order embedded runge – kutta formulae, J. comput. appl. math., 7, 1, 67-75, (1981) · Zbl 0449.65048
[16] Rabe, E., Determination and survey of periodic trojan orbits in the restricted problem of three bodies, Astronom. J., 66, 9, 500-513, (1961)
[17] Shampine, L.F.; Gordon, M.K., Computer solution of ordinary differential equations: the initial value problem, (1975), Freeman San Francisco, CA · Zbl 0347.65001
[18] Sharp, P.W., Numerical comparison of explicit runge – kutta pairs of orders four through eight, Trans. math. software, 17, 387-409, (1991) · Zbl 0900.65236
[19] Szebehely, V., Theory of orbits, the restricted problem of three bodies, (1967), Acad. Press New York · Zbl 0202.56902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.