NINE-STAGE MULTI-DERIVATIVE RUNGE-KUTTA METHOD OF ORDER 12

A nine-stage multi-derivative Runge-Kutta method of order 12, called HBT(12) 9, is constructed for solving nonstiff systems of first-order differential equations of the form y' = f(x, y), y(x(0)) = y(0). The method uses y' and higher derivatives y((2)) to y((6)) as in Taylor methods and is combined with a 9-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The stepsize is controlled by means of the derivatives y((3)) to y((6)). The new method has a larger interval of absolute stability than Dormand-Prince's DP(8,7) 13M and is superior to DP(8,7) 13M and Taylor method of order 12 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, maximum global error of position and energy. Numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.