Lagmental Vicfred

Classical RK4 Blends Four Slopes by Vicfred

tags: mathematics, ordinary differential equations, Runge--Kutta methods, stability

Four carefully weighted stage evaluations yield fourth-order global accuracy. A small computation will anchor the general statement before the abstraction takes over.

The mathematical object

An initial-value problem \(y'=f(t,y)\), \(y(t_0)=y_0\) generates a flow when existence and uniqueness hold. A numerical method advances discrete states \(y_n\approx y(t_n)\).

$$ k_1=f(t_n,y_n),\quad k_2=f(t_n+\tfrac h2,y_n+\tfrac h2k_1) $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4) $$

One explicit computation

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \begin{aligned}k_3&=f(t_n+\tfrac h2,y_n+\tfrac h2k_2),\\k_4&=f(t_n+h,y_n+hk_3),\\y(t_n)-y_n&=O(h^4).\end{aligned} $$

Why the identity matters

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad k_1=f(t_n,y_n),\quad k_2=f(t_n+\tfrac h2,y_n+\tfrac h2k_1),\\[5pt] \mathsf{C}\;&:\quad y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4). \end{aligned} $$

Where it can fail

Order of accuracy and stability region are separate. A high-order explicit method can fail spectacularly on a stiff equation.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] y_{n+1}=y_n+\frac h6(k_1+2k_2+2k_3+k_4) \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Tue 19 January 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.