NINE - STAGE MULTI - DERIVATIVE RUNGE – KUTTA METHOD OF ORDER 12 Truong Nguyen - Ba

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(x0) = y0. 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.