Can you explain me how does the backward Euler method works? I have seen the formula and try to understand the method, but what I can't understand is why and how to use the Newton-Rapson method.
Do you have a link to a good tutorial? Something that can help me understand it graphically?
UPDATE
According to what Paul said, could you please tell me if what follows is correct?
The Cauchy problem is like this
$$u(t) = f\big ( u(t), t \big ) \quad \text{where} \quad f\big ( u(t), t \big ) = u'\big ( u(t), t \big )$$
then, according to the example given by Paul, I'll have something like:
$$u'(t)=\big [u(t) \big]^3$$
Now, the backward Euler method reads:
$$u_{t+1} = u_t + hf\big ( u_{t+1}, t+1 )~~~~~~~~~~\text{not really sure about this}$$
So for, $t = 0$ and $h = 1/2$ I should have the following
$$u_1 = u_0 + \frac{1}{2}(u_1)^3$$
For this equation I have to solve by means of Newton-Raphson before going to the next time $t = 1$:
$$u_1 - u_0 - \frac{1}{2}(u_1)^3 = 0$$
From here I don't know what to do, how to use Newton's method??
UPDATE
I think I finally got it. From backward Euler's method I have:
$$u_{t+1} = u_t + hf(u_{t+1}, t+1)$$
this can be seen as (just like everybody told me):
$$ F(u_{t+1})= u_{t+1} - u_t - hf(u_{t+1}, t+1) = 0$$
where $u_{t+1} = x$ so I have
$$ F(x)= x - u_t - hf(x, t+1) = 0$$
which now I see a familiar way to use Newton's (or whatever other method) to find the value of $x$