0
$\begingroup$

Given the modified Euler method: $u_{n+1} = u_n + hf(u_n + \frac{h}{2}f(u_n))$ applied to the test equation $y' = f(y) = \lambda y$, how do you prove that no imaginary value $h\lambda$ is contained in the region of absolute stability?

I've found that the this region is $-2 \le h\lambda \le 0$, but how is it that no imaginary values are contained in that region?

$\endgroup$
0
$\begingroup$

Actually, the region is given by $$ |1+z+\frac{z^2}2|<1,\qquad z=λh, $$ which indeed on the real axis reduces to the interval $[-2,0]$.

For $z=iy$ one finds $$ |1-\frac{y^2}2+iy|<1\iff (1-\frac{y^2}2)^2+y^2<1\iff y^4<0 $$ which clearly is impossible.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.