Bogacki-Shampine method

Explicit, three-stage, third-order, variable-step, Runge-Kutta method.¹

$\frac{d y}{d t} = f (t, y)$ $t_{n + 1} = t_{n} + h$ $k_{1} = f (t_{n}, y_{n})$ $k_{2} = f (t_{n} + \frac{1}{2} h, y_{n} + \frac{1}{2} h k_{1})$ $k_{3} = f (t_{n} + \frac{3}{4} h, y_{n} + \frac{3}{4} h k_{2})$ $y_{n + 1} = y_{n} + \frac{h}{9} (2 k_{1} + 3 k_{2} + 4 k_{3})$ $k_{4} = f (t_{n + 1}, y_{n + 1})$ $e_{n + 1} = \frac{h}{72} (- 5 k_{1} + 6 k_{2} + 8 k_{3} - 9 k_{4})$

P. Bogacki and L.F. Shampine, Appl. Math. Lett. 2, 321 (1989). ↩

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Bogacki-Shampine method