Euler's Method

A careful guide to Euler's numerical method, stepwise computation, error sources, and responsible approximation.

Euler’s method advances from one approximate point using the current local slope. A sequence of indexed points moves from an initial value toward later approximations. Each step uses a fixed horizontal increment while the displayed output values change according to the local slope.

Precise definition

For the initial value problem $y'=f(t,y)$, $y(t_0)=y_0$, Euler's method with step $h$ defines $t_{n+1}=t_n+h$ and $y_{n+1}=y_n+h f(t_n,y_n)$. It follows the tangent line from the current approximation for one step; the values $y_n$ approximate, rather than equal, $y(t_n)$.

Notation and mathematical language

The local truncation error of forward Euler is $O(h^2)$ per step when the exact solution is sufficiently smooth, while accumulated global error over a fixed interval is $O(h)$. This order statement is asymptotic and does not supply a numerical bound without derivative and stability information.

Conceptual picture

Euler replaces a curved solution over each short interval by its tangent at the left endpoint. Smaller steps usually improve accuracy, but rounding, stiffness, and instability can complicate the pattern. The method is explicit because $y_{n+1}$ is computed directly from known data.

Fully worked example

Interpretation and application

Euler's method approximates models that lack elementary closed forms and previews more accurate numerical solvers. In safety-critical simulation, a small step is not a proof of accuracy: an error tolerance, convergence study, and suitable method for stiffness are required.