= Crank-Nicolson centered-advection scheme on a finite interval
{c}
Let $D$ be the real skew-symmetric tridiagonal matrix with superdiagonal $1$ and subdiagonal $-1$. The centered-space Crank-Nicolson discretization of $u_t=u_x$ has amplification matrix
$$
Q=(I-\mu D/4)^{-1}(I+\mu D/4).
$$
The eigenvalues of $D$ are $2i\cos(j\pi/(M+1))$, so those of $Q$ are Cayley transforms
$$
q_j=\frac{1+i(\mu/2)\cos(j\pi/(M+1))}
{1-i(\mu/2)\cos(j\pi/(M+1))}.
$$
They all have modulus one, and their common orthonormal eigenbasis makes $Q$ normal. Hence the method is stable for every $\mu>0$.
Back to article page
