DormandPrince#
- class brainpy.integrators.ode.DormandPrince(f, var_type=None, dt=None, name=None, adaptive=None, tol=None, show_code=False, state_delays=None, neutral_delays=None)[source]#
The Dormand–Prince method for ODEs.
The DOPRI method, is an explicit method for solving ordinary differential equations (Dormand & Prince 1980). The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the FSAL (First Same As Last) 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 (Shampine 1986; Hairer, Nørsett & Wanner 2008, pp. 178–179).
Its Butcher table is:
\[\begin{split}\begin{array}{l|llllll} 0 & \\ 1 / 5 & 1 / 5 & & & \\ 3 / 10 & 3 / 40 & 9 / 40 & & & \\ 4 / 5 & 44 / 45 & -56 / 15 & 32 / 9 & & \\ 8 / 9 & 19372 / 6561 & -25360 / 2187 & 64448 / 6561 & -212 / 729 & \\ 1 & 9017 / 3168 & -355 / 33 & 46732 / 5247 & 49 / 176 & -5103 / 18656 & \\ 1 & 35 / 384 & 0 & 500 / 1113 & 125 / 192 & -2187 / 6784 & 11 / 84 & \\ \hline & 35 / 384 & 0 & 500 / 1113 & 125 / 192 & -2187 / 6784 & 11 / 84 & 0 \\ & 5179 / 57600 & 0 & 7571 / 16695 & 393 / 640 & -92097 / 339200 & 187 / 2100 & 1 / 40 \end{array}\end{split}\]References