A family of embedded Runge-Kutta formulae
Citations
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3,330 citations
Cites methods from "A family of embedded Runge-Kutta fo..."
...The new ode23 is based on the Bogacki-Shampine (2, 3) pair [3] (see also [37]) and the new ode45 is based on the Dormand-Prince (4, 5) pair [12]....
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1,349 citations
Cites methods from "A family of embedded Runge-Kutta fo..."
...For example, the ode45 routine in MATLAB uses a pair of embedded Runge-Kutta methods of order 4 and 5 due to Dormand and Prince [25]....
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1,174 citations
Cites methods from "A family of embedded Runge-Kutta fo..."
...Using a black-box ODE solver (Dormand & Prince, 1980) not only produces high quality samples (Table 2, details in Appendix D.4), but also allows us to explicitly trade-off accuracy for efficiency....
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...In our experiments, we use the RK45 ODE solver (Dormand & Prince, 1980) provided by scipy.integrate.solve_ivp in all cases....
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673 citations
Cites background or methods from "A family of embedded Runge-Kutta fo..."
...It has been suggested by Dormand and Prince [5] that every elementary differential should contribute to both the local error and its estimate....
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...With this in mind, we chose the coefficients of our CSIRK formulas so that their local truncation errors have the form suggested by Dormand and Prince [5] and Shampine [13]....
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References
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"A family of embedded Runge-Kutta fo..." refers background or methods in this paper
...the two "minimum truncation error" formulae RK5(4)7M, RK5 (4)6M are more efficient than the RKF4 (Fehlberg [8])....
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...Fehlberg ([7], [8]) has developed embedded RK formulae which have a 'small' principal truncation term in the lower order formula....
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...Practical tests [8], however, indicate that, in spite of our comments in section 1, good results for the lower order mode are obtained by attempting to make the a i as small as possible....
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...For RKF4 [8], which practical results ([5], [6]) indicate is preferable in local extrapolation mode, 11&(6) 112 = 3....
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394 citations
"A family of embedded Runge-Kutta fo..." refers background in this paper
...nr + 1' being the elementary differentials [1] of order r + i offNote that if the formula is of order p, then -Or --- 0, r = 1, 2 ....
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...Butcher [1] has listed the expressions for the elementary differentials for up to order 8 and Harris [3] has considered the computer derivation of the equations of condition....
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...Without loss of generality [1] the first order autonomous system y_'(x) = f [y (x)], (1....
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346 citations
"A family of embedded Runge-Kutta fo..." refers methods in this paper
...Fehlberg ([7], [8]) has developed embedded RK formulae which have a 'small' principal truncation term in the lower order formula....
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...Shampine [15] has developed a modification to the RKF7 formula of Fehlberg [7] which overcomes this difficulty but it is preferable to develop formulae which are free from this deficiency....
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...Concerning the quadrature problem, it has been noted previously [ 14] that many embedded formulae of high order, such as those of Fehlberg [7] with p ;~ 5, and Dormand & Prince [11] fail because, in this case, the two formulae yield the same numerical approximation for Y(Xn + 1)....
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