One-step 5-stage Hermite-Birkhoff-Taylor ODE solver of order 12

A one-step 4-stage Hermite-Birkhoff-Taylor method of order 12, denoted by HBT(12)4, is constructed for solving nonstiff systems of first-order differential equations of the form y′ = f(x, y), y(x0) = y0. The method uses derivatives y′ to y(9) as in Taylor methods combined with a 4-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to Taylorand Runge-Kutta-type order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than DormandPrince DP(8,7)13M and 4-stage Hermite-Birkhoff-Obrechkoff method of order 12. The stepsize is controlled by a formula which uses two high-order derivatives. HBT(12)4 is superior to DP(8,7)13M and Taylor method of order 12 in solving several test problems on the basis of the number of steps, CPU time, and maximum global error, thus showing the benefit of adding high-order derivatives to Runge-Kutta methods.