Implicit Scheme
The implicit scheme is always stable since the spectral radius satisfies \( \rho(\mathbf{A}) < 1 \). We could have inferred this by noting that
the matrix is positive definite, viz. all eigenvalues are larger than zero. We see this from
the fact that \( \mathbf{A}=\hat{I}+\alpha\hat{B} \) has eigenvalues
\( \lambda_i = 1+\alpha(2-2cos(\theta)) \) which satisfy \( \lambda_i > 1 \). Since it is the inverse which stands
to the right of our iterative equation, we have \( \rho(\mathbf{A}^{-1}) < 1 \)
and the method is stable for all combinations
of \( \Delta t \) and \( \Delta x \).