Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 41A (Numerical Analysis)

The exact average is $I(h)={\hat{h}}_0$. For one Fourier mode,Thus the periodic trapezoidal Fourier aliasing identity givesUnder the coefficient bound,This decays exponentially in $N$.

WriteProjecting the equation onto modes $|n|\le D$ gives the Fourier-Galerkin matrix for a drift-diffusion equationwhere ${\hat{w}}_l=0$ for $|l|>d$.

For $w=\cos \pi x$, only
${\hat{w}}_{\pm 1}=1/2$ are nonzero. HenceThe column belonging to the constant mode $m=0$ is zero, soRemove that zero mode. In every remaining row $n$, the sum of the magnitudes of the off-diagonal entries is at most ${\pi }^2|n|$; for $|n|=1$, the would-be coupling to the removed zero mode vanishes. The diagonal entry is $-{\pi }^2{n}^2$. Gershgorin's theorem therefore places every eigenvalue of the nonconstant block inTogether with the zero eigenvalue, all eigenvalues of $B$ have nonpositive real part.