A one-step 7-stage Hermite–Birkhoff–Taylor ODE solver of order 11

Abstract

A one-step 7-stage Hermite–Birkhoff–Taylor method of order 11, denoted by HBT(11)7, is constructed for solving nonstiff first-order initial value problems y ′ = f ( t , y ) , y ( t 0 ) = y 0 . The method adds the derivatives y ′ to y ( 6 ) , used in Taylor methods, to a 7-stage Runge–Kutta method of order 6. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution to order 11 leads to Taylor- and Runge–Kutta-type order conditions. These conditions 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 the Dormand–Prince DP87 and a larger unscaled interval of absolute stability than the Taylor method, T11, of order 11. HBT(11)7 is superior to DP87 and T11 in solving several problems often used to test higher-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. Numerical results show the benefit of adding high-order derivatives to Runge–Kutta methods.