ODE Solver Methods

Watch Euler, Heun (RK2), and RK4 race the true solution of an ODE. Push the step size and see which methods keep up — and which fall apart.

ODE Solver MethodsLive

Euler vs Heun vs RK4

Controls

Integrate exact eˣ. Raise the step size to watch Euler drift (or, on the stiff ODE, blow up) while RK4 stays glued to the analytic curve.

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Presets

Step size h

x-range end

▶ Run in Python

Data Inspector

ODEy' = y

Steps8

Euler error2.90e+1

Heun error5.98e+0

RK4 error7.51e-2

Order (E / H / RK4)0.60 / 1.69 / 3.70

Governing equation

k_{1} k_{2} k_{3} k_{4} y_{n + 1} Euler: = f (x_{n}, y_{n}) = f (x_{n} + \frac{h}{2}, y_{n} + \frac{h}{2} k_{1}) = f (x_{n} + \frac{h}{2}, y_{n} + \frac{h}{2} k_{2}) = f (x_{n} + h, y_{n} + h k_{3}) = y_{n} + \frac{h}{6} (k_{1} + 2 k_{2} + 2 k_{3} + k_{4}) y_{n + 1} = y_{n} + h f (x_{n}, y_{n}), h = 0.50

Reading this result: Each step commits a local truncation error — O(h²) for Euler, O(h³) for Heun, O(h⁵) for RK4 — and those accumulate over ~8 steps into the global error shown (one order lower: O(h), O(h²), O(h⁴)). That extra power of h is why halving the step cuts Euler's error ~2×, Heun's ~4×, and RK4's ~16×, so RK4 hugs the true y(x) far more tightly for the same work.

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How it works

Each method advances dy/dx = f(x, y) forward in fixed steps of size h, but they differ in how they estimate the slope over each step. Euler uses one slope evaluation (O(h) global error), Heun averages two (O(h²)), and RK4 blends four (O(h⁴)). Overlaid against the known analytic solution, the gap is obvious: at large h Euler drifts or — on a stiff ODE — blows up entirely, while RK4 stays glued to the true curve. Halving the step cuts Euler's error roughly 2×, Heun's 4×, and RK4's 16×, the payoff of higher-order accuracy that makes RK4 the workhorse of numerical ODE solving.

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Frequently asked questions

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How accurate are the results?▾

The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.