Explicit Euler Method

The explicit Euler method (also called forward Euler) is the simplest first-order numerical method for solving ODEs of the form $\dot{y}=g(t,y)$.

$$y_{i+1}=y_i+dt\cdot g(t_i,y_i)$$

It uses the forward difference approximation for the derivative, so all quantities on the right-hand side are known at step $i$ - making it fully explicit.

Order

The explicit Euler method is first-order: Global truncation error $O(dt)$, Local truncation error $O(d{t}^2)$.

Stability

For the test ODE $\dot{y}=-ay$ (with $a>0$), the method gives

$$y_{i+1}=(1-a\cdot dt)y_i$$

The amplification factor is $|1-a\cdot dt|$. The method is unstable when $|1-a\cdot dt|>1$, i.e. when

$$a\cdot dt>2$$

So the explicit Euler method is conditionally stable - there is a maximum allowable step size. Compare with the Implicit Euler method, which is unconditionally stable.

See Stability of a method for the general framework.

Implicit Euler method | Finite differences | Runge-Kutta methods | Local truncation error | Global truncation error | Stability of a method | Order of a method