A one-step 7-stage Hermite–Birkhoff–Taylor ODE solver of order 11

Nguyen‐Ba, Truong; Bozic, Vladan; Kengne, Emmanuel; Vaillancourt, Rémi

doi:10.1016/j.cam.2009.12.015

Cited by 4 publications

(5 citation statements)

References 21 publications

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“…The considered problems are Kepler's, Hénon-Heiles' and the equatorial main problems over the time interval [0, t f ] Table 1. For some test problems of [11], time interval [0,20] and LT = log 10 (TOL), the table lists the number of steps (NS) and the maximum global error (GE) for HBT(12)9 (left column) and T12 (right column).…”

Section: Numerical Results Related To the Step Controlmentioning

confidence: 99%

“…The exponents, in the above formula, come from 1/7 = (11/7)(1/11) and 1/8 = (12/8)(1/12). It was observed that HBT(12)9 solves the ODEs considered in this paper more efficiently with stepsize h n+1 obtained by (22) without rejected steps than by means of a step control predictor. In (22), η acts as control factor in the variable step algorithm.…”

Section: 2mentioning

confidence: 87%

“…It was observed that HBT(12)9 solves the ODEs considered in this paper more efficiently with stepsize h n+1 obtained by (22) without rejected steps than by means of a step control predictor. In (22), η acts as control factor in the variable step algorithm. If η is set to 1.0 as the assumption C PET = y (13) /13!, the stepsize of HBT(12)9 is very conservative.…”

Section: 2mentioning

confidence: 87%

“…then η > 1. This result will be used to justify the value of the factor η in the stepsize formula (22) in the next subsection.…”

Section: Region Of Absolute Stabilitymentioning

confidence: 99%

“…5 (right)). Since HBT (12)9 does not use derivatives of order higher than six, to determine the stepsize we shall use the following formula (22) …”

Section: 2mentioning

confidence: 99%

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

Nguyen‐Ba

Bozic

Kengne

et al. 2009

Publ Inst Math (Belgr)

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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.

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Section: Numerical Results Related To the Step Controlmentioning

confidence: 99%

Section: 2mentioning

confidence: 87%