Extensão: Limites e erros pelo metodo de Euler

When approximating the initial value problem $y' = f(t, y)$ using Euler's Method (of the first order), given that $f$ is Lipschitz continuous and satisfies $|y''(t)|\le M$ for some $M$, it is possible to establish a global error bound. In most cases, $f$ is Lipschitz continuous with Lipschitz constant $L$ if $\left| \frac{\partial f}{\partial y} \right| \le L$ holds for all $(t,y)$ in the domain of the problem.

We'll begin by finding an expression for $y(t_{i+1})$ in terms of $y(t_i)$.

Right. The $\xi_i$ might seem a little strange, but we'll deal with it later. Let's clean things up a little, using $h=t_{i+1}-t_i$ (the step size) to get $y(t_{i+1})=y(t_i)+h\cdot f(t_i,y(t_i))+h^2/2 \cdot y''(\xi_i)$. Additionally, for convenience, let $w_i$ be our approximation to $y(t_i)$, and let $y_i=y(t_i)$ (the real value).

By definition, we have $w_{i+1} = w_i + h\cdot f(t_i, w_i)$. Subtract this from the previous result to obtain:

$$y_{i+1} - w_{i+1} = y_i - w_i + h (f(t_i,y_i) - f(t_i, w_i)) + h^2/2 \cdot y''(\xi_i)$$

In an attempt to find a bound on the maximum absolute error, we take the absolute value of both sides and apply both the triangle inequality and our assumption about $|y''|$ above:

$$\left| y_{i+1} - w_{i+1} \right| \le \left| y_i - w_i \right| + h \left| f(t_i,y_i) - f(t_i, w_i) \right| + h^2M/2$$

We now arrive at the bound $\left| y_{i+1} - w_{i+1} \right| \le (1+hL) \left| y_i - w_i \right| + h^2M/2$. Notice the recurrent geometric relationship. We'll establish some useful lemmas now to help simplify this down:

Lemma 1: $0 \le (1+x)^m \le e^{mx}$ for all $x \ge -1$ and positive $m$

Now exponentiate by $m$ for the desired result. Onwards!

Lemma 2: For positive $s, t$, $a_0 \ge -t/s$, and $a_{i+1} \le (1+s) a_i + t$, we have: $$a_{i+1} \le e^{(i+1)s} \left( a_0 + \frac{t}{s} \right) - \frac{t}{s}$$

Proof: Compute the sum of the finite geometric series and apply the previous lemma.

Finally, then, we have our final conclusion. With $s=hL$, $t=h^2M/2$, and $a_0 = y_0-w_0 = 0$, we can simplify our previous recursive error bound to the following nicer result: $$\left| y_{i+1} - w_{i+1} \right| \le \frac{hM}{2L} \left( e^{L(t_{i+1}-t_0)} - 1 \right)$$

Note that this bound is linear in $h$. Higher order Euler methods will yield similarly higher order error bounds. One should take care when attempting to drive $h$ too small, as computational floating point error may begin to dominate the previously derived approximation error.