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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((i1)θ)=μisin(iθ), or 2(1cos(θ))sin(iθ)=μisin(iθ), which is nothing but 2(1cos(θ))xi=μixi, with eigenvalues μi=22cos(θ).

Our requirement in Eq. (8) results in 1<1α2(1cos(θ))<1, which is satisfied only if α<(1cos(θ))1 resulting in α1/2 or Δt/Δx21/2.