Order of Consistency for Euler Method

Euler Method

The Euler method is defined by $y_{i+1}=y_i+hf(t_i,y_i)$. The increment function is $\Phi (t,y,h)=f(t,y)$.

1. Truncation error :

   $$\tau (t,h)=\frac{y(t+h)-y(t)}{h}-\Phi (t,y(t),h)$$

2. Use Taylor Series: We expand $y(t+h)$ around $t$:

   $$y(t+h)=y(t)+h{y}^{\prime }(t)+\mathcal{O}(h^2)$$

   Note: Comparing the expansion with the given rule, we get $y^{\prime }=f$

3. Substitute and Simplify:

   $$\tau (t,h)=\frac{\left[y(t)+h{y}^{\prime }(t)+\mathcal{O}(h^2)\right]-y(t)}{h}-f(t,y(t))$$ $$\tau (t,h)=y^{\prime }(t)+\mathcal{O}(h)-f(t,y(t))$$

   Since $y^{\prime }(t)=f(t,y(t))$, the leading terms cancel:

   $$\tau (t,h)=\mathcal{O}(h)$$

The local truncation error is of the first order in $h$. Therefore, the Euler method has an order of consistency of 1.