Learning high-order spatial discretisations of PDEs with symmetry-preserving iterative algorithms

The paper proves from identity (32a) that truncating to o(γ^p) yields a stencil of width 2p−1 and consistency o(∂_x^p) for the general diffusion-like PDE (27); its argument pairs powers of γ with δ and expands asinh^2, establishing exact recovery of ∂_x^2 at γ=1 and the stencil/consistency bounds (Theorem 25) . The candidate solution presents a closely related but more explicit series/degree-counting proof: it rewrites (32a) as asinh^2 of Z=(γ/2)δ√(A/B), expands in powers of Z, bounds the δ-degree for each γ-degree, shows A=B when γ=1 so the operator is exactly ∂_x^2, and then shows K(·)=∑K_k(·)^k preserves the lowest derivative order. Minor differences (e.g., an unnecessary mention of “K1 invertible”) do not affect correctness. Hence both are correct, with substantively similar structure but different emphases.