Fourier spectral method for variable-coefficient advection

For $u_t+c(x)u_x=0$ with two-periodic Fourier truncations $|n|\le d$, let $C_{nm}={\hat{c}}_{n-m}$ and $D_{mm}=m$. Fourier projection gives

If a real Fourier polynomial $c(x)$ is strictly positive, then $C_{nm}={\hat{c}}_{n-m}$ is Hermitian positive definite becausefor every nonzero $z$.

If $C$ is Hermitian positive definite and $D$ is Hermitian, every eigenvalue of $CD$ is real. Equivalently, $CD$ is similar to the Hermitian matrix $C^{1/2}D{C}^{1/2}$.

For $y^{\prime }=i\omega y$ with real nonzero $\omega$, explicit Euler has amplification factor $1+i\omega h$, whose modulus $\sqrt{1+{\omega }^2{h}^2}$ exceeds one for every $h>0$.