In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations (ODE).[1] The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth-order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Cash–Karp (RKCK).
The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the "First Same As Last" (FSAL) property: the last stage is evaluated at the same point as the first stage of the next step. Dormand and Prince chose the coefficients of their method to minimize the error of the fifth-order solution. This is the main difference with the Fehlberg method, which was constructed so that the fourth-order solution has a small error. For this reason, the Dormand–Prince method is more suitable when the higher-order solution is used to continue the integration, a practice known as local extrapolation.[2][3]
The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order accurate solution.
Applicationsedit
As of 2023[update], Dormand–Prince is the default method in the ode45 solver for MATLAB[4] and GNU Octave[5] and is the default choice for the Simulink's model explorer solver. It is an option in Python's SciPy ODE integration library[6] and in Julia (programming language)'s ODE solvers library.[7] Implementations for the languages Fortran,[8]Java,[9] and C++[10] are also available.
Notesedit
^Dormand, J.R.; Prince, P.J. (1980). "A family of embedded Runge-Kutta formulae". Journal of Computational and Applied Mathematics. 6 (1): 19–26. doi:10.1016/0771-050X(80)90013-3.
^Shampine, Lawrence F. (1986). "Some Practical Runge-Kutta Formulas". Mathematics of Computation. 46 (173): 135–150. doi:10.2307/2008219. JSTOR 2008219.
^Hairer, Ernst; Wanner, Gerhard; Nørsett, Syvert P. (1993). Solving Ordinary Differential Equations I. Springer Series in Computational Mathematics. Vol. 8. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-540-78862-1. ISBN978-3-540-56670-0.
^"Solve nonstiff differential equations — medium order method - MATLAB ode45". www.mathworks.com. Retrieved 2023-08-24.
Dormand, John R. (1996), Numerical Methods for Differential Equations: A Computational Approach, Boca Raton: CRC Press, pp. 82–84, ISBN0-8493-9433-3
Further readingedit
Articlesedit
C. Engstler, C. Lubich (1997), "MUR8: a multirate extension of the eighth-order Dormand-Prince method", Applied Numerical Mathematics, vol. 25, no. 2–3, pp. 185–192, doi:10.1016/S0168-9274(97)00058-5
M. Calvo, J.I. Montijano, L. Randez (1990), "A fifth-order interpolant for the Dormand and Prince Runge-Kutta method", Journal of Computational and Applied Mathematics, vol. 29, no. 1, pp. 91–100, doi:10.1016/0377-0427(90)90198-9{{citation}}: CS1 maint: multiple names: authors list (link)
Jeffrey M. Aristoff, Joshua T. Horwood, Aubrey B. Poore (2014), "Orbit and uncertainty propagation: a comparison of Gauss–Legendre-, Dormand–Prince-, and Chebyshev–Picard-based approaches", Celestial Mechanics and Dynamical Astronomy, vol. 118, no. 1, pp. 13–28, Bibcode:2014CeMDA.118...13A, doi:10.1007/s10569-013-9522-7, S2CID 254378950{{citation}}: CS1 maint: multiple names: authors list (link)
Wo Mei Seen, R. U. Gobithaasan, Kenjiro T. Miura (2014), "GPU acceleration of Runge Kutta-Fehlberg and its comparison with Dormand-Prince method", Germination of Mathematical Sciences Education and Research Towards Global Sustainability (Sksm21), AIP Conference Proceedings, Penang, Malaysia, 1605 (1): 16–21, Bibcode:2014AIPC.1605...16S, doi:10.1063/1.4887558{{citation}}: CS1 maint: multiple names: authors list (link)
December 03, 2023
dormand, prince, method, numerical, analysis, dormand, prince, rkdp, method, dopri, method, embedded, method, solving, ordinary, differential, equations, method, member, runge, kutta, family, solvers, more, specifically, uses, function, evaluations, calculate,. In numerical analysis the Dormand Prince RKDP method or DOPRI method is an embedded method for solving ordinary differential equations ODE 1 The method is a member of the Runge Kutta family of ODE solvers More specifically it uses six function evaluations to calculate fourth and fifth order accurate solutions The difference between these solutions is then taken to be the error of the fourth order solution This error estimate is very convenient for adaptive stepsize integration algorithms Other similar integration methods are Fehlberg RKF and Cash Karp RKCK The Dormand Prince method has seven stages but it uses only six function evaluations per step because it has the First Same As Last FSAL property the last stage is evaluated at the same point as the first stage of the next step Dormand and Prince chose the coefficients of their method to minimize the error of the fifth order solution This is the main difference with the Fehlberg method which was constructed so that the fourth order solution has a small error For this reason the Dormand Prince method is more suitable when the higher order solution is used to continue the integration a practice known as local extrapolation 2 3 Contents 1 Butcher tableau 2 Applications 3 Notes 4 References 4 1 Books 5 Further reading 5 1 ArticlesButcher tableau editThe Butcher tableau is 01 5 1 53 10 3 40 9 404 5 44 45 56 15 32 98 9 19372 6561 25360 2187 64448 6561 212 7291 9017 3168 355 33 46732 5247 49 176 5103 186561 35 384 0 500 1113 125 192 2187 6784 11 8435 384 0 500 1113 125 192 2187 6784 11 84 05179 57600 0 7571 16695 393 640 92097 339200 187 2100 1 40The first row of b coefficients gives the fifth order accurate solution and the second row gives the fourth order accurate solution Applications editAs of 2023 update Dormand Prince is the default method in the ode45 solver for MATLAB 4 and GNU Octave 5 and is the default choice for the Simulink s model explorer solver It is an option in Python s SciPy ODE integration library 6 and in Julia programming language s ODE solvers library 7 Implementations for the languages Fortran 8 Java 9 and C 10 are also available Notes edit Dormand J R Prince P J 1980 A family of embedded Runge Kutta formulae Journal of Computational and Applied Mathematics 6 1 19 26 doi 10 1016 0771 050X 80 90013 3 Shampine Lawrence F 1986 Some Practical Runge Kutta Formulas Mathematics of Computation 46 173 135 150 doi 10 2307 2008219 JSTOR 2008219 Hairer Ernst Wanner Gerhard Norsett Syvert P 1993 Solving Ordinary Differential Equations I Springer Series in Computational Mathematics Vol 8 Berlin Heidelberg Springer Berlin Heidelberg doi 10 1007 978 3 540 78862 1 ISBN 978 3 540 56670 0 Solve nonstiff differential equations medium order method MATLAB ode45 www mathworks com Retrieved 2023 08 24 Matlab compatible solvers GNU Octave version 8 3 0 octave org Retrieved 2023 08 24 scipy integrate ode SciPy v1 11 2 Manual docs scipy org Retrieved 2023 08 24 ODE Solvers DifferentialEquations jl docs sciml ai Retrieved 2023 08 24 Hairer Ernst Fortran Codes www unige ch Retrieved 2023 08 24 DormandPrince54Integrator Apache Commons Math 4 0 beta1 commons apache org Retrieved 2023 08 24 Class template runge kutta dopri5 1 53 0 www boost org Retrieved 2023 08 24 References editBooks edit Dormand John R 1996 Numerical Methods for Differential Equations A Computational Approach Boca Raton CRC Press pp 82 84 ISBN 0 8493 9433 3Further reading editArticles edit C Engstler C Lubich 1997 MUR8 a multirate extension of the eighth order Dormand Prince method Applied Numerical Mathematics vol 25 no 2 3 pp 185 192 doi 10 1016 S0168 9274 97 00058 5 M Calvo J I Montijano L Randez 1990 A fifth order interpolant for the Dormand and Prince Runge Kutta method Journal of Computational and Applied Mathematics vol 29 no 1 pp 91 100 doi 10 1016 0377 0427 90 90198 9 a href Template Citation html title Template Citation citation a CS1 maint multiple names authors list link Jeffrey M Aristoff Joshua T Horwood Aubrey B Poore 2014 Orbit and uncertainty propagation a comparison of Gauss Legendre Dormand Prince and Chebyshev Picard based approaches Celestial Mechanics and Dynamical Astronomy vol 118 no 1 pp 13 28 Bibcode 2014CeMDA 118 13A doi 10 1007 s10569 013 9522 7 S2CID 254378950 a href Template Citation html title Template Citation citation a CS1 maint multiple names authors list link Wo Mei Seen R U Gobithaasan Kenjiro T Miura 2014 GPU acceleration of Runge Kutta Fehlberg and its comparison with Dormand Prince method Germination of Mathematical Sciences Education and Research Towards Global Sustainability Sksm21 AIP Conference Proceedings Penang Malaysia 1605 1 16 21 Bibcode 2014AIPC 1605 16S doi 10 1063 1 4887558 a href Template Citation html title Template Citation citation a CS1 maint multiple names authors list link Retrieved from https en wikipedia org w index php title Dormand Prince method amp oldid 1183964336, wikipedia, wiki, book, books, library,
