Abstract
The new variable-step, variable-order, ODE solver, HBT( p) of order p, presented in this paper, combines a three-stage Runge–Kutta method of order 3 with a Taylor series method of order p - 2 to solve initial value problems y ′ = f ( t , y ) , y ( t 0 ) = y 0 , where y : R → R d and f : R × R d → R d . The order conditions satisfied by HBT( p) are formulated and they lead to Vandermonde-type linear algebraic systems whose solutions are the coefficients in the formulae for HBT( p). A detailed formulation of variable-step HBT( p) in both fixed-order and variable-order modes is presented. The new method and the Taylor series method have similar regions of absolute stability. To obtain high-accuracy results at high order, this method has been implemented in multiple precision.
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