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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 .