Dormand–Prince method

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Short description: Method for solving differential equations

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]

Butcher tableau

The Butcher tableau is:

0
1/5 1/5
3/10 3/40 9/40
4/5 44/45 −56/15 32/9
8/9 19372/6561 −25360/2187 64448/6561 −212/729
1 9017/3168 −355/33 46732/5247 49/176 −5103/18656
1 35/384 0 500/1113 125/192 −2187/6784 11/84
35/384 0 500/1113 125/192 −2187/6784 11/84 0
5179/57600 0 7571/16695 393/640 −92097/339200 187/2100 1/40

The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order accurate solution.

Applications

(As of 2023), 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

  1. ↑ Dormand, J.R.; Prince, P.J. (1980). "A family of embedded Runge-Kutta formulae" (in en). Journal of Computational and Applied Mathematics 6 (1): 19–26. doi:10.1016/0771-050X(80)90013-3. https://linkinghub.elsevier.com/retrieve/pii/0771050X80900133. 
  2. ↑ Shampine, Lawrence F. (1986). "Some Practical Runge-Kutta Formulas". Mathematics of Computation 46 (173): 135–150. doi:10.2307/2008219. https://www.jstor.org/stable/2008219. 
  3. ↑ Hairer, Ernst; Wanner, Gerhard; Nørsett, Syvert P. (1993) (in en). Solving Ordinary Differential Equations I. Springer Series in Computational Mathematics. 8. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-540-78862-1. ISBN 978-3-540-56670-0. http://link.springer.com/10.1007/978-3-540-78862-1. 
  4. ↑ "Solve nonstiff differential equations — medium order method - MATLAB ode45". https://www.mathworks.com/help/matlab/ref/ode45.html. 
  5. ↑ "Matlab-compatible solvers (GNU Octave (version 8.3.0))". https://octave.org/doc/interpreter/Matlab_002dcompatible-solvers.html#Matlab_002dcompatible-solvers. 
  6. ↑ "scipy.integrate.ode — SciPy v1.11.2 Manual". https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.ode.html. 
  7. ↑ "ODE Solvers · DifferentialEquations.jl". https://docs.sciml.ai/DiffEqDocs/stable/solvers/ode_solve/. 
  8. ↑ Hairer, Ernst. "Fortran Codes". http://www.unige.ch/~hairer/software.html. 
  9. ↑ "DormandPrince54Integrator (Apache Commons Math 4.0-beta1)". https://commons.apache.org/proper/commons-math/javadocs/api-4.0-beta1/org/apache/commons/math4/legacy/ode/nonstiff/DormandPrince54Integrator.html. 
  10. ↑ "Class template runge_kutta_dopri5 - 1.53.0". https://www.boost.org/doc/libs/1_53_0/libs/numeric/odeint/doc/html/boost/numeric/odeint/runge_kutta_dopri5.html. 

References

Books

  • Dormand, John R. (1996), Numerical Methods for Differential Equations: A Computational Approach, Boca Raton: CRC Press, pp. 82–84, ISBN 0-8493-9433-3 

Further reading

Articles

  • C. Engstler, C. Lubich (1997), "MUR8: a multirate extension of the eighth-order Dormand-Prince method", Applied Numerical Mathematics 25 (2–3): 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 29 (1): 91–100, doi:10.1016/0377-0427(90)90198-9 
  • 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 118 (1): 13–28, doi:10.1007/s10569-013-9522-7, Bibcode: 2014CeMDA.118...13A 
  • 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, doi:10.1063/1.4887558, Bibcode: 2014AIPC.1605...16S 



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