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

Truong Nguyen-Ba, Vladan Bozic, Emmanuel Kengne, and Rémi Vaillancourt

Abstract: 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.

Keywords: general linear method for non-stiff ODE; Hermite–Birkhoff method; Taylor method; maximum global error; number of function evaluations; CPU time

Classification (MSC2000): 65L06; 65D05; 65D30

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