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Volume 14,     Number 1,     Spring 2006

 

VARIABLE-STEP VARIABLE-ORDER 3-STAGE
HERMITE-BIRKHOFF ODE SOLVER OF
ORDER 5 TO 15
TRUONG NGUYEN-BA, HEMZA YAGOUB, YI LI
AND RÉMI VAILLANCOURT

Abstract. Variable-step variable-order 3-stage Hermite- Birkhoff (HB) methods HB(p)3 of order p = 5 to 15 are constructed& for solving non-stiff differential equations. 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 confluent Vandermonde-type systems of HB type. Fast algorithms are developed for solving these systems in O(p2) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stability regions of the HB methods have a remarkably good shape. The order and stepsize of these methods are controlled by four local error estimators. When programmed in C++, HB(p)3 uses less CPU time than Dormand-Prince DP(8,7)13M in solving costly problems at stringent tolerance.

 

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© 2006, Canadian Applied Mathematics Quarterly (CAMQ)