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Implicit Scheme

The implicit scheme is always stable since the spectral radius satisfies ρ(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 A=ˆI+αˆB has eigenvalues λi=1+α(22cos(θ)) which satisfy λi>1. Since it is the inverse which stands to the right of our iterative equation, we have ρ(A1)<1 and the method is stable for all combinations of Δt and Δx.