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(2−2cos(θ)) which satisfy
λi>1. Since it is the inverse which stands
to the right of our iterative equation, we have
ρ(A−1)<1
and the method is stable for all combinations
of
Δt and
Δx.