# Classical Runge-Kutta Method

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## 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

This method is just one member of the family of Runge-Kutta Methods and so is often referred to with more specific names:

**RK4****Classical Runge-Kutta Method**

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 - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Runge-Kutta method**(C.D.T. Runge, 1895; W.M. Kutta, 1901)