Euler's Method

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.

Conditions and key results

$f$ should be evaluable throughout the numerical path, and an exact solution should exist over the interval. Error estimates require additional smoothness. A step crossing a singularity or leaving the model's domain invalidates the computation even if arithmetic continues.

A reliable strategy

1. Choose $h$, create columns for $n,t_n,y_n,f(t_n,y_n)$, and record the initial row.
2. Evaluate the slope using the current pair only, then compute $y_{n+1}=y_n+h f(t_n,y_n)$.
3. Advance $t$ by exactly $h$ and repeat for the required number of steps.
4. Report the result as an approximation and assess error by step halving, an exact solution, or a rigorous bound.

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.

Common mistakes

Verification and reasonableness

• Repeat with $h/2$ and compare values at common time points.
• When an exact solution exists, compute absolute and relative error.
• Check qualitative properties—sign, monotonicity, feasible range—against the differential equation.

Practice

1. Take one Euler step for $y'=t+y$, $y(0)=2$, $h=0.1$.
2. Is Euler's $y_n$ exact?
3. What is forward Euler's global order?

Further deduction

For $y'=y$, Euler gives $y_n=(1+h)^n$ at $t_n=nh$, whereas the exact solution is $e^{nh}$. Since $1+h<e^h$ for $h>0$, Euler underestimates this growing solution at every positive grid time. This exact comparison explains the sign of the error in the worked example and illustrates how a qualitative error claim can sometimes be proved rather than guessed from one table.

Related topics

• Direction Fields
• Initial Value Problems
• Existence and Uniqueness