Classical Runge-Kutta Method

Proof Technique

Consider the first order ODE:

$(1): \quad y' = \map f {x, y}$ subject to the initial condition $\map y {x_0} = y_0$

where $\map f {x, y}$ is continuous.

Let $\map y x$ be the particular solution of $(1)$.

For all $n \in \N_{>0}$, we define:

$x_n = x_{n - 1} + h$

where $h \in \R_{>0}$.

Let the following numbers be calculated:

$\ds m_1$

$=$

$\ds h \map f {x_n, y_n}$

$\ds m_2$

$=$

$\ds h \map f {x_n + \frac h 2, y_n + \frac {m_1} 2}$

$\ds m_3$

$=$

$\ds h \map f {x_n + \frac h 2, y_n + \frac {m_2} 2}$

$\ds m_4$

$=$

$\ds h \map f {x_n + h, y_n + m_3}$

Then for all $n \in \N_{>0}$ such that $x_n$ is in the domain of $y$:

$y_{n + 1} = y_n + \dfrac 1 6 \paren {m_1 + 2 m_2 + 2 m_3 + m_4}$

is an approximation to $\map y {x_{n + 1} }$.

Proof

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Also known as

The method is also referred to as:

the four-stage Runge-Kutta method

RK4

but usually it is simply known as the Runge-Kutta method.

Source of Name

This entry was named for Carl David Tolmé Runge and Martin Wilhelm Kutta.

Sources

1972: George F. Simmons: Differential Equations ... (previous) ... (next): Miscellaneous Problems for Chapter $2$: Appendix $\text{A}$. Numerical Methods
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Runge-Kutta method (C.D.T. Runge, 1895; W.M. Kutta, 1901)
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Runge-Kutta method (C.D.T. Runge, 1895; W.M. Kutta, 1901)