In numerical analysis, the Runge–Kutta methods (English: RUUNG-ə-KUUT-tah) are a family of explicit and implicit iterative methods, which include the Euler method, used in discretization for the approximate solutions of differential equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta.
The fourth-order Runge–Kutta method
The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows:
d y d t = f ( t , y ) , y ( t 0 ) = y 0 . {\displaystyle {\frac {dy}{dt}}=f(t,y),\quad y(t_{0})=y_{0}.}
Here y {\displaystyle y} is an unknown function (scalar or vector) of time t {\displaystyle t} , which we would like to approximate; we are told that d y d t {\displaystyle {\frac {dy}{dt}}} , the rate at which y {\displaystyle y} changes, is a function of t {\displaystyle t} and of y {\displaystyle y} itself. At the initial time t 0 {\displaystyle t_{0}} the corresponding y {\displaystyle y} value is y 0 {\displaystyle y_{0}} . The function f {\displaystyle f} and the initial conditions t 0 {\displaystyle t_{0}} , y 0 {\displaystyle y_{0}} are given. Now we pick a step-size h > 0 and define:
y n + 1 = y n + h 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) , t n + 1 = t n + h {\displaystyle {\begin{aligned}y_{n+1}&=y_{n}+{\frac {h}{6}}\left(k_{1}+2k_{2}+2k_{3}+k_{4}\right),\\t_{n+1}&=t_{n}+h\\\end{aligned}}}
for n = 0, 1, 2, 3, ..., using
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