A three-stage, VSVO, Hermite–Birkhoff–Taylor, ODE solver

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-2p-2 to solve initial value problems y′=f(t,y),y(t0)=y0, where y:R→Rdy:R→Rd and f:R×Rd→Rdf:R×Rd→Rd. 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.