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Volume 13,     Number 2,     Summer 2005

 

HERMITE-BIRKHOFF-OBRECHKOFF 3-STAGE
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.

 

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© 2005, Canadian Applied Mathematics Quarterly (CAMQ)