6.2 Runge–Kutta Methods
Probably one of the more popular, as well as most accurate, numerical procedures used in obtaining approximate solutions to a first-order initial-value problem y′ = f(x, y), y(x0) = y0 is the fourth-order Runge–Kutta method devised by the German applied mathematicians Carl David Runge (1856–1927) and Martin Wilhelm Kutta (1867–1944) in the late 1890s. As the words fourth-order suggest, there are Runge–Kutta methods of different orders.
Fundamentally, all Runge–Kutta methods are generalizations of the basic Euler formula (1) of Section 6.1 in that the slope function f is replaced by a ...
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