Implicit Runge–Kutta method

In numerical analysis, the Runge–Kutta methods are a family of implicit and explicit iterative methods, which includes the well-known routine called the Euler Method, used in temporal discretization for the approximate solutions of ordinary differential equations. These methods were developed around 1900 by the German mathematicians C. Runge and M. W. Kutta.

See the article on numerical methods for ordinary differential equations for more background and other methods. See also List of Runge–Kutta methods.

The most widely known member of the Runge–Kutta family is generally referred to as "RK4", "classical Runge–Kutta method" or simply as "the Runge–Kutta method".

Let an initial value problem be specified as follows:

Here y is an unknown function (scalar or vector) of time t, which we would like to approximate; we are told that ${\displaystyle {\dot {y}}}$, the rate at which y changes, is a function of t and of y itself. At the initial time ${\displaystyle t_{0}}$ the corresponding y value is ${\displaystyle y_{0}}$. The function f and the data ${\displaystyle t_{0}}$, ${\displaystyle y_{0}}$ are given.

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