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 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:

\(\displaystyle m_1\) \(=\) \(\displaystyle h \map f {x_n, y_n}\)
\(\displaystyle m_2\) \(=\) \(\displaystyle h \map f {x_n + \frac h 2, y_n + \frac {m_1} 2}\)
\(\displaystyle m_3\) \(=\) \(\displaystyle h \map f {x_n + \frac h 2, y_n + \frac {m_2} 2}\)
\(\displaystyle m_4\) \(=\) \(\displaystyle h \map f {x_n + h, y_n + m_3}\)


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


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