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

Abstract

A one-step 5-stage Hermite–Birkhoff–Taylor method, HBT(12)5, of order 12 is constructed for solving nonstiff systems of differential equations y ′ = f ( t , y ) , y ( t 0 ) = y 0 , where y ∈ R n . The method uses derivatives y ′ to y ( 9 ) as in Taylor methods combined with a 5-stage Runge–Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution to order 12 leads to Taylor- and Runge–Kutta-type order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. HBT(12)5 has a larger interval of absolute stability than Dormand–Prince DP(8, 7)13M and Taylor method T12 of order 12. The new method has also a smaller norm of principal error term than T12. It is superior to DP(8, 7)13M and T12 on the basis the number of steps, CPU time and maximum global error on common test problems. The formulae of HBT(12)5 are listed in an appendix.