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 .