Euler Method

Definition

Euler’s method is the simplest numerical scheme for solving an initial-value problem. Given $\frac{dy}{dx}=f(x,y)$ with $y(x_0)=y_0$, a single step of size $\Delta x$ is

$$y_{n+1}=y_n+\Delta xf(x_n,y_n).$$

The tangent-line slope at $(x_n,y_n)$ is used to march the solution forward.

Intuition

Euler’s method is simply the repeated application of the first-order Taylor approximation: the solution curve is replaced by its tangent line at each step. The smaller the step $\Delta x$, the less the tangent departs from the true curve before the next correction. The accumulated error grows as $O(\Delta x)$ — one power of $\Delta x$ is “spent” on the approximation at each step, giving a first-order method.

Formal Description

First-order ODE

Repeat the step $y_{n+1}=y_n+\Delta xf(x_n,y_n)$ until the desired $x$ is reached. For small enough $\Delta x$ the numerical solution converges to the exact solution.

Higher-order ODEs

A second-order ODE $\ddot{x}=f(t,x,\dot{x})$ is first reduced to a first-order system by setting $u=\dot{x}$:

$$\dot{x}=u,\dot{u}=f(t,x,u).$$

Starting from $(x_0,u_0)$ at $t_0$, each step advances both variables simultaneously:

$$x_{n+1}=x_n+\Delta t{u}_n,u_{n+1}=u_n+\Delta tf(t_n,x_n,u_n).$$

The same reduction applies to ODEs of any order: introduce one new variable per derivative to obtain a first-order system.

Key Results

• Euler’s method is a first-order method: the global error is $O(\Delta t)$.
• The method is simple but generally less accurate than higher-order methods such as Runge-Kutta.
• Any $n$th-order ODE can be converted to a system of $n$ first-order ODEs and then integrated numerically.

Applications

• Quick prototyping and teaching of numerical ODE concepts.
• Baseline reference against which higher-order methods are compared.
• Embedded in more sophisticated adaptive solvers as the predictor step.

Trade-offs

• First-order accuracy means halving the step size only halves the error; achieving high accuracy requires very small $\Delta t$ and many steps.
• Can be numerically unstable for stiff ODEs unless the step size is very small.
• Runge-Kutta methods achieve much higher accuracy with the same number of function evaluations.

Links

• Runge-Kutta Methods
• Linear First-Order ODE