Euler method

Summary

The Euler method is a numerical method to approximate the solution of an initial value problem $y' = f(t, y)$, $y(t_0) = y_0$. With a step size $h > 0$, it sets $t_{k+1} = t_k + h$ and $y_{k+1} = y_k + h\, f(t_k, y_k)$ for $k = 0, 1, 2, \ldots$. Then $y_k$ approximates $y(t_k)$. It follows the tangent line of the solution for one step at a time.

Remarks

The error of one step, the local truncation error, is $\frac{h^2}{2} y''(\tau)$ for some $\tau$ between $t_k$ and $t_{k+1}$, when $y$ is twice continuously differentiable. Let $f$ be $L$-Lipschitz in $y$ with $L > 0$, and let $|y''| \le M$ on the interval. Then the global error satisfies $|y(t_k) - y_k| \le \frac{h M}{2 L} \left( e^{L (t_k - t_0)} - 1 \right)$. The proof uses a discrete Grönwall inequality. So the method is of first order: halving the step size roughly halves the error. It is the simplest Runge–Kutta method. Leonhard Euler proposed it in his book Institutionum calculi integralis.

Explanations

L1 human

At each step, we compute the slope at the current point and walk a short distance $h$ along it. The result is a broken line that stays close to the true solution when $h$ is small. For $y' = y$, $y(0) = 1$, the method gives $y_k = (1 + h)^k$, and with $h = 1/k$ this tends to $e$ at time $1$.