Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 41A (Numerical Analysis)

Expanding the square and using orthogonality of Fourier modes,Finite truncations justify the calculation directly, and passage to the ${\ell }^2$ limit proves Parseval identity for every square-summable sequence.

Fourier transformation in $m$ turns the recurrence intoAn amplification factor $z$ therefore satisfiesFor $\mu >0$, the quadratic Schur stability criterion giveswhere $a_1=-2\mu \cos \theta /(1+\mu )$ and $a_2=(\mu -1)/(\mu +1)$. Thus both roots lie in the closed unit disk; roots on it at $\theta =0,\pi$ are simple. At $\mu =0$, the roots are $\pm 1$. Parseval's identity then gives stability precisely forwhich is the full stated parameter range.

The matrix exponential is the absolutely convergent seriesMultiplying the two expansions givesOn the other hand,Subtracting yieldsThis is the leading Lie-Trotter splitting commutator error.

Expanding the numerical propagation matrix,Its difference from $e^{kA}$ isTherefore the one-step local truncation error isand the method is first-order accurate globally.

If $BC=CB$, the splitting commutator vanishes, but the backward-Euler resolvents still contributeThus commutativity alone does not change the order: the local error remains $O(k^2)$ and the global method remains first order, barring an additional special cancellation.