Foundations — Runge-Kutta 4th order (RK4) — derivation

This page builds every piece of vocabulary the parent note RK4 derivation leans on — starting from things a 12-year-old already knows and adding exactly one new idea at a time. Nothing here is assumed; if a symbol appears, it was defined a line earlier.

1. A function $f$ and the notation $f(x,y)$

The letter $x$ is the horizontal position (how far right we've walked). The letter $y$ is the vertical position (the height). So $f(x,y)$ is a rule that, given where you are on a map (a right-distance and a height), hands you back a single number.

In our topic that returned number is a slope — how tilted the path is at that spot. Look at Figure 1: at every point of the plane, $f$ paints a little arrow showing which way to lean.

2. Slope, and the symbol $y^{\prime }$

Picture a ramp. If for every $1$ metre you walk right you rise $2$ metres, the slope is $2$. We give this "rate of rising" a shorthand symbol:

So the whole problem of our topic is a sentence: $y^{\prime }=f(x,y)$ which reads: "the slope of my curve, at any point, is given by the machine $f$." We know the slope-rule everywhere; we do not know the curve itself. Our job is to reconstruct the curve.

3. The Initial Value Problem (IVP) and $y(x_0)=y_0$

Knowing the slope everywhere still isn't enough — infinitely many curves have the same slope-field, just shifted up or down. We must nail down one starting point.

A slope-rule plus a starting point is called an Initial Value Problem (IVP). Figure 1's arrows are the slope-rule; a single dot pins the curve. From that dot, the arrows tell you where to go next.

4. Subscripts $n$, $n+1$ — marching in steps

We can't draw the curve in one smooth swoop; we take it one small hop at a time.

So $y_0\to y_1\to y_2\to \cdots$ is a chain of stepping-stones across the fog.

5. The step size $h$

Figure 3 shows the same step done with small vs large $h$: the small step hugs the true curve, the large step can drift.

6. The integral ${\int }_{x_n}^{x_n+h}fdx$

The parent note writes the exact answer as an integral. Here is that symbol from zero.

So the exact step is "new height = old height + total rise": $y(x_n+h)=y_n+{\int }_{x_n}^{x_n+h}f(x,y(x))dx.$

7. Big-O notation: $O(h^5)$, $O(h^4)$

• Local error $O(h^5)$: the mistake made in one single step.
• Global error $O(h^4)$: the total mistake after crossing a whole interval. Crossing takes about $1/h$ steps, so per-step $O(h^5)$ errors pile up into $O(h^4)$ — one power is lost. (More in Local vs Global Truncation Error.)

8. Taylor series — the "true answer" we chase

The parent note tunes RK4 to match the Taylor series of the real curve. What is that?

Here $y^{\prime \prime }$ ("y double-prime") is the slope of the slope — how fast the tilt itself is changing (the curve's bending). RK4's cleverness: it reproduces those terms without ever computing $y^{\prime \prime },y^{\prime \prime \prime },\dots$ — it fakes them with extra evaluations of $f$. See Taylor Series Methods for the version that computes them directly.

9. The stage slopes $k_1,k_2,k_3,k_4$

You don't need to derive these here — the parent note does. You only need to read $k_i$ as "a walked-height guess from one peek at the slope." Modified Euler / Heun's Method uses just two peeks (edges); RK4 uses four.

Prerequisite map

Equipment checklist

Read the prompt, answer in your head, then reveal.

Connections

• Euler's Method — uses only $k_1$ (one left-edge peek).
• Modified Euler / Heun's Method — two peeks (both edges).
• Simpson's Rule — source of the $1,2,2,1$ blending weights.
• Taylor Series Methods — computes $y^{\prime \prime },y^{\prime \prime \prime }$ directly; RK4 fakes them.
• Local vs Global Truncation Error — the $O(h^5)$ vs $O(h^4)$ story.
• Adaptive Step Size (RKF45) — chooses $h$ automatically.