====== DOPRI45 ======

Dormand-Prince method 

In numerical analysis, the Dormand–Prince method or DOPRI method, is an embedded method for solving ordinary differential equations (Dormand & Prince 1980). 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. 

Applicable for **non-stiff** problems of **medium** accuracy.

The [[Butcher_tableau|Butcher tableau]] for the adaptive Dormand–Prince method 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.


Applicable [[Solver_settings|solver settings]]:

  * Absolute tolerance
  * Relative tolerance
  * Initial step size
  * Max step size
  * Min step size
  * Refine
  * Limit data points to last
  * Norm Control
  * Allowed step size violations
  * Enable saturation

===== Reference =====

  * Dormand, J. R. and P. J. Prince, “A family of embedded Runge-Kutta formulae,” //J. Comp. Appl. Math.//, Vol. 6, 1980, pp 19-26.

  * http://en.wikipedia.org/wiki/Dormand-Prince