Fourier-Galerkin matrix for a drift-diffusion equation
= Fourier-Galerkin matrix for a drift-diffusion equation
{c}
For $u_t=u_{xx}-w'u_x$ and Fourier truncation $|n|\leq D$,
$$
\dot{\widehat u}_n=\sum_{|m|\leq D}B_{nm}\widehat u_m,
$$
where
$$
B_{nm}=-\pi^2n^2\delta_{nm}
+\pi^2(n-m)m\,\widehat w_{n-m}.
$$
For $w(x)=\cos\pi x$, the nonconstant block has Gershgorin discs in the closed left half-plane, while the constant mode lies in the kernel. Hence every eigenvalue has nonpositive real part and the matrix is singular.