Publications de l'Institut Mathématique, Nouvelle Série Vol. 86(100), pp. 75–96 (2009) 

NINESTAGE MULTIDERIVATIVE RUNGE–KUTTA METHOD OF ORDER 12Truong NguyenBa, Vladan Bozic, Emmanuel Kengne, and Rémi VaillancourtDepartment of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, CanadaAbstract: A ninestage multiderivative Runge–Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of firstorder 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 Vandermondetype 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 highorder 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 highorder derivatives to Runge–Kutta methods. Keywords: general linear method for nonstiff ODE; Hermite–Birkhoff method; Taylor method; maximum global error; number of function evaluations; CPU time Classification (MSC2000): 65L06; 65D05; 65D30 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 4 Nov 2009. This page was last modified: 26 Nov 2009.
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