Volume 13, Number 2, Summer 2005
6-STEP ODE SOLVER OF ORDER 14
TRUONG NGUYEN-BA AND RÉMI VAILLANCOURT
Abstract. A 3-stage 6-step variable step
Hermite-Birkhoff-Obrechkoff method of order 14, denoted by HBO14(3,6),
is constructed for solving non-stiff systems of first-order differential
equations of the form y' = ∫(x,y), y(x0) = y0. Its
formula uses y' and y" as in Obrechkoff method. 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. Fast algorithms are developed for
solving these systems to obtain Hermite-Birkhoff interpolation
polynomials in terms of generalized Lagrange basis functions.
The new method has a larger region of absolute stability than
Adams-Bashforth-Moulton methods of orders 10 to 13 in PECE
mode. The stepsize is controlled by a local error estimator.
HBO14(3,6) is superior to MATLAB's ode113 in solving several
problems often used to test higher order ode solvers based on
the number of steps, cpu time, and maximum global error.