Geodesic
Nine-stage Multi-derivative Runge-Kutta method of order 12
Publications de l'Institut Mathématique, _N_S_86 (2009) no. 100, p. 75 .

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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.
DOI : 10.2298/PIM0900075N
Classification : 65L06 65D05 65D30
Keywords: general linear method for non-stiff ODE, Hermite-Birkhoff method, Taylor method, maximum global error, number of function evaluations, CPU time 
@article{10_2298_PIM0900075N,
     author = {Truong Nguyen-Ba and Vladan Bo\v{z}i\'c and Emmanuel Kengne and R\'emi Vaillancourt},
     title = {Nine-stage {Multi-derivative} {Runge-Kutta} method of order 12},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {75 },
     publisher = {mathdoc},
     volume = {_N_S_86},
     number = {100},
     year = {2009},
     doi = {10.2298/PIM0900075N},
     zbl = {1265.65134},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/PIM0900075N/}
}
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Truong Nguyen-Ba; Vladan Božić; Emmanuel Kengne; Rémi Vaillancourt. Nine-stage Multi-derivative Runge-Kutta method of order 12. Publications de l'Institut Mathématique, _N_S_86 (2009) no. 100, p. 75 . doi : 10.2298/PIM0900075N. http://geodesic.mathdoc.fr/articles/10.2298/PIM0900075N/

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