# Backwards Euler Method/ Implicit Euler Scheme - Difference Equation + Stability

The backwards Euler method (implicit Euler scheme) is a numerical method for the finding the solution of ordinary differential equations, which is defined as follows, $$y(t_{n+1}) \approx y(t_n) + hf(t_{n+1}, y(t_{n+1}))$$ where $$h = t_{n+1} - t_n$$.

How could one find and solve the difference equation resulting from using $$f(y) =-y \,$$ for this method? What does this then say about the unconditional stability of the backward Euler method?

• Right out the first few steps explicitly and express $y_(t_j)$ as a function of $j$, $h$, and $y(t_0)$ for $j=1,2,3$. A pattern should emerge that you can then formally prove. Commented Apr 26, 2021 at 9:03
• @CarlChristian Thanks for the comment. I'm not too sure what you mean by this. Could you format an answer so I could see? Commented Apr 28, 2021 at 9:53
• So you get $$y_{n+1}=y_n+h·(-y_{n+1})$$ and want to know how $y_{n+1}$ computes from $y_n$,... Commented May 2, 2021 at 11:19

We have $$y(t_{n+1})=y(t_n)-hy(t_{n+1})$$
$$(1+h)y(t_{n+1})=y(t_n)$$
$$y(t_{n+1}) = \frac1{1+h}y(t_n)$$
$$y(t_{n})=\left(\frac1{1+h} \right)^n y(t_0)$$
Note that we have $$0<\frac1{1+h}<1$$ for $$h>0$$, it is unconditionally stable.