We consider the following initial value problem:
$ y'(t) = \begin{pmatrix}
-y_{2}(t) \\
y_1(t)\\
\end{pmatrix} $ with $ y(0) = \begin{pmatrix}
a \\
0\\
\end{pmatrix} $
with $ a \in \mathbb{R} $ and $ ( y_1(t), y_2(t))^T \in \mathbb{R^2} $.
Determine the exact solution and calculate $ \lim_{t \rightarrow \infty} ||y(t)||_2^2 $.
Then consider the Explicit Euler Method and the Implicit Euler Method. $(y_k)_{k \in \mathbb{N}}$ is sequence generated by using the Explicit Euler Method / Implicit Euler Method.
Show:
$ \lim_{k \rightarrow \infty} ||y_k||_2^2 = \infty $ ( for the explicit method ) and $ \lim_{k \rightarrow \infty} ||y_k||_2^2 = 0 $ ( for the implicit method ) .
Approach:
Ok, so If Im not mistaken the exact solution has to be:
$ y(t) = \begin{pmatrix}
a* \cos(t) \\
a * \sin(t)\\
\end{pmatrix} $ okey then I calculate $ \lim_{t \rightarrow \infty} ||y(t)||_2^2 $
$ \lim_{t \rightarrow \infty} ||y(t)||_2^2 = \lim_{t \rightarrow \infty} (a*\cos(t))^2 + (a*\sin(t))^2 = a^2$ Hope theres no mistake.
Then the part where I stuck is the explicit/implicit part.
I know the definition of these two methods:
Explicit Euler Method: $ y_{k+1} = y_{k} + h*f(t_k,y_k) $
Implicit Euler Method: $ y_{k+1} = y_{k} + h*f(t_{k+1},y_{k+1}) $
So my problem is to "find" the sequence $(y_k)_{k \in \mathbb{N}}$ after using these methods infinite number of times.
Therefore I can't calculate $lim_{k \rightarrow \infty} ||y_k||_2^2$.
Can you help me out here?
Thank you in advance.
Best Answer
The easiest way is to recognize that the system matrix represents an imaginary unit, so with $z=y_1+iy_2$ you get $z'=iz$ and in the Euler methods $z_{n+1}=(1+ih)z_n$ for the explicit method and $z_{n+1}=(1-ih)^{-1}z_n$ for the implicit methods. The result immediately follows.
If you want to avoid this special property, the equation is still linear, and so are the Euler methods applied to it. So determine the system matrix of an integration step and determine its eigenvalues and if the all lie inside or all outside of the unit circle.