Dormand-Prince method

Explicit, six-stage, fifth-order, variable-step, Runge-Kutta method.¹

$\frac{d y}{d t} = f (t, y)$ $t_{n + 1} = t_{n} + h$ $k_{1} = f (t_{n}, y_{n})$ $k_{2} = f (t_{n} + \frac{1}{5} h, y_{n} + \frac{1}{5} h k_{1})$ $k_{3} = f (t_{n} + \frac{3}{10} h, y_{n} + \frac{3}{40} h k_{1} + \frac{9}{40} h k_{2})$ $k_{4} = f (t_{n} + \frac{4}{5} h, y_{n} + \frac{44}{45} h k_{1} - \frac{56}{15} h k_{2} + \frac{32}{9} h k_{3})$ $k_{5} = f (t_{n} + \frac{8}{9} h, y_{n} + \frac{19372}{6561} h k_{1} - \frac{25360}{2187} h k_{2} + \frac{64448}{6561} h k_{3} - \frac{212}{729} h k_{4})$ $k_{6} = f (t_{n} + h, y_{n} + \frac{9017}{3168} h k_{1} - \frac{355}{33} h k_{2} - \frac{46732}{5247} h k_{3} + \frac{49}{176} h k_{4} - \frac{5103}{18656} h k_{5})$ $y_{n + 1} = y_{n} + h (\frac{35}{384} k_{1} + \frac{500}{1113} k_{3} + \frac{125}{192} k_{4} - \frac{2187}{6784} k_{5} + \frac{11}{84} k_{6})$ $k_{7} = f (t_{n + 1}, y_{n + 1})$ $e_{n + 1} = \frac{h}{5} (\frac{71}{11520} k_{1} - \frac{71}{3339} k_{3} + \frac{71}{384} k_{4} - \frac{17253}{67840} k_{5} + \frac{22}{105} k_{6} - \frac{1}{8} k_{7})$

J.R. Dormand and P.J. Prince, J. Comput. Appl. Math. 6, 19 (1980). ↩

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Dormand-Prince method