Explicit Scheme, stability condition
If we assume that
x can be expanded in a basis of
x=(sin(θ),sin(2θ),…,sin(nθ))
with
θ=lπ/n+1, where we have the endpoints given by
x0=0 and
xn+1=0, we can rewrite the
last equation as
2sin(iθ)−sin((i+1)θ)−sin((i−1)θ)=μisin(iθ),
or
2(1−cos(θ))sin(iθ)=μisin(iθ),
which is nothing but
2(1−cos(θ))xi=μixi,
with eigenvalues
μi=2−2cos(θ).
Our requirement in
Eq. (8) results in
−1<1−α2(1−cos(θ))<1,
which is satisfied only if α<(1−cos(θ))−1 resulting in
α≤1/2 or Δt/Δx2≤1/2.