Euler integration

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In mathematics and computational science, Euler integration (or the Euler method) is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic kind of explicit numerical integration for ordinary differential equations.

[edit] Derivation

We want to approximate the solution of the initial value problem

$y'(t) = f(t,y(t)), \qquad \qquad y(t_0)=y_0,$

by using the first two terms of the Taylor expansion of y. One step of Euler Integration from t_n to t_n+1 = t_n + h is

$y_{n+1} = y_n + hf(t_n,y_n). \qquad \qquad$

The Euler method of integration is explicit, i.e. the solution $y n + 1$ is an explicit function of $y i$ for $i \leq n$ .

While Euler integration integrates a first order ODE, any ODE of order $N$ can be represented as a first-order ODE in more than one variable by introducing $N - 1$ further variables, $y'$ , $y''$ , ..., $y (N)$ , and formulating $N$ first order equations in these new variables. The Euler method can be applied to the vector $\mathbf{y}(t)=(y(t),y'(t),y''(t),...,y^{(N)}(t))$ to integrate the higher-order system.

[edit] Error

The magnitude of the errors arising from Euler integration can be demonstrated by comparison with a Taylor expansion of y. If we assume that f(t) and y(t) are known exactly at a time t₀, then Euler integration gives the approximate solution at time t₀ + h as:

$y(t_0 + h) = y(t_0) + h f(t_0,y(t_0)). \, \qquad \qquad$

In comparison, the Taylor expansion in $h$ about $t 0$ gives:

$y(t_0 + h) = y(t_0) + h y'(t_0) + \frac{1}{2}h^2 y''(t_0) + O(h^3).$

The error introduced by Euler integration is given by the difference between these equations:

$-\frac{1}{2}h^2 y''(t_0) + O(h^3).$

For small $h$ , the dominant error per step is proportional to $h 2$ . To solve the problem over a given range of $t$ , the number of steps needed is proportional to $1 / h$ so it is to be expected that the total error at the end of the fixed time will be proportional to $h$ (error per step times number of steps). For this reason, Euler's method is said to be first order. This makes Euler integration less accurate (for small $h$ ) than other higher-order techniques such as Runge-Kutta methods and linear multistep methods.

Euler integration can also be numerically unstable, especially for stiff equations. This limitation—along with its slow convergence of error with $h$ —means that Euler integration is not often used, except as a simple example of numerical integration.