The midpoint method, also known as the second-order Runga-Kutta method, improves the Euler method by adding a midpoint in the step which increases the accuracy by one order. The midpoint method is not the only second-order Runge–Kutta method with two stages. In fact, there is a family of such methods, parameterized by α, and given by the formula [ 8 ]

The statement "fourth-order Runge-Kutta is generally superior to second-order" is a true one, but you should recognize it as a statement about the 712 Chapter 16.Oct 21, 2011 · The work of Runge was extended by (Heun 1900), who completed a discussion of order 3 methods and pointed the way to order 4, and by (Kutta 1901) who gave a complete classification of order 4 methods. The most well-known method, due to Runge, has order 4 and is defined by the tableau