Nine-stage Multi-derivative Runge-Kutta method of order 12

Truong,Nguyen-Ba, Vladan,Božić, Emmanuel,Kengne, Rémi,Vaillancourt

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 methodsand 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 solutionleads to order conditions which are reorganized into Vandermonde-type linear systemswhose 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)13Mand is superior to DP(8,7)13M and Taylor method of order 12 in solving several problemsoften 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.