I am trying to prove that when solving numerically diff. eq.:
$$
y'(t)=f(t,y(t)), \hspace{0.5cm} y(t_{0})=y_{0}
$$
using trapezoidal rule, namely:
$$
y_{n+1}=y_{n} + \frac{h}{2} \left( f(t_{n},y_{n}) + f(t_{n+1},y_{n+1}) \right)
$$
we get LTE (local truncation error) proportional to $C h^{3}$, $C=const$.
Let's denote LTE in $n+1$-th step as:
$$
e_{n+1}= \phi(t_{n+1}) – y_{n+1}
$$
where $\phi(t_{n})$ represents the actual solution of our problem at $t=t_{n}$ and we assume that at the $n$-th step our approximating solution is exact with the real one ($\phi(t_{n}) = y_{n}$).
I know that the most common method starts at representing $\phi(t)$ as the Taylor series about $t_{n+1}=t_{n}+h$ and using the fact that $\phi'(t_{n}) = f(t_{n},\phi(t_{n}))$. In this particular example I used two Taylor representations: $\phi(t_{n+1})=\phi(t_{n}+h)$ and $\phi(t_{n+1}-h)=\phi(t_{n})$. Combining them resulted in the formula:
$$
\phi(t_{n+1})=\phi(t_{n}) + \frac{h}{2}(\phi'(t_{n})+\phi'(t_{n+1})) + \frac{h^{2}}{4}(\phi''(t_{n})-\phi''(t_{n+1})) + \frac{h^{3}}{12}(\phi'''(\xi_{n})+\phi'''(\eta_{n+1}))
$$
where:
$$
t_{n}<\xi_{n}<t_{n}+h
$$
$$
t_{n}<\eta_{n+1}<t_{n}+h
$$
So plugging this obtained formula for $\phi(t_{n+1})$ into the equation
$$
e_{n+1} = \phi(t_{n+1}) – y_{n+1} = \phi(t_{n+1}) -y_{n} – \frac{h}{2} \left( f(t_{n},y_{n}) + f(t_{n+1},y_{n+1}) \right)
$$
and assuming that $\phi(t_{n})=y_{n}$ resulted in:
$$
e_{n+1} = \frac{h^{2}}{4}(\phi''(t_{n})-\phi''(t_{n+1})) + \frac{h^{3}}{12}(\phi'''(\xi_{n})+\phi'''(\eta_{n+1}))
$$
So it would be $e_{n+1}=O(h^{2})$, not $O(h^{3})$ (and we know that it is $O(h^{3})$). I saw in the literature that assuming correctness of approximation at $t_{n}$ means that trapezoidal rule can be represented as:
$$
y_{n+1} = y_{n} + \frac{h}{2} \left( f(t_{n},\phi(t_{n})) + f(t_{n+1},\phi(t_{n+1})) \right)
$$
but it doesn't make sense to me because of the correctness of the term $\phi(t_{n+1})$.
I hope that I presented my problem clearly, thanks in advance for any help with solving it!
Best Answer
you are almost there: The term $\phi''(t_{n})-\phi''(t_{n+1})$ is $ h * \phi'''(\eta)$ for some $t_n \le \eta \le t_{n+1}$
The trapzoidal is symmtery at $t_n + \frac{1}{2}h$. The elegant way to do it is to expand $\phi(t_n) $ and $\phi(t_{n+1})$ at $t_n + \frac{1}{2}h$.