Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 41C / b / Solution

For $u_t+c(x)u_x=0$ with two-periodic Fourier truncations $|n|\leq d$, let $C_{nm}=\widehat c_{n-m}$ and $D_{mm}=m$. Fourier projection gives
$$
\dot{\widehat{\mathbf u}}=-i\pi CD\widehat{\mathbf u}.
$$


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 Incoming links: Fourier spectral method for variable-coefficient advection