Polytropic Dynamical Systems with Time Singularity

Both aim to prove uniform O(ĥ) global error for forward Euler on the mesh ĥ = h_{n+1}/(1−t_{n+1}) (equivalently t_{n+1} = t_n + ĥ(1−t_{n+1})). The paper sets up integral remainders and derives a discrete inequality, then states Theorem 4.3 with the desired O(ĥ) global error. However, there are internal inconsistencies in the paper’s proof: (i) after defining the mesh by ĥ = h_{n+1}/(1−t_{n+1}) (equation (19)), it later asserts (1−ĥ)^N = δ to relate N and δ, which corresponds to a different mesh (one based on t_n, not t_{n+1}) and is inconsistent with (19); the correct relation from (19) is 1−t_N = (1+ĥ)^{−N} (so N ≈ ln(1/δ)/ĥ as ĥ→0), not (1−ĥ)^N (see the mesh definition and Theorem 4.3 statement around equation (19) and its proof) . (ii) In applying the discrete Grönwall lemma, the paper appears to set M1 = 1 + m1(ĥ) when the recurrence already has the factor (1 + m1(ĥ)), effectively double-counting the “+1” and leading to an exponential factor e^{(N+1)M1} that is too large; the simplification from inequality (22) to the Grönwall-ready form also introduces a notational anomaly “m2(ĥ)^3” (dimensions inconsistent with standard local-error-squared scaling) . These issues are fixable, but as written the proof is not internally coherent at key steps. By contrast, the model’s solution gives a standard, self-contained proof: it converts to first order, establishes a uniform Lipschitz constant on [0,1−δ], derives O(ĥ^2) local defect via Taylor bounds on x'', x''', and applies discrete Grönwall to obtain uniform O(ĥ) global error. The mesh is solved exactly (t_n = 1−(1+ĥ)^{−n}), the step-sum identities are correct, and the bootstrap closes. Hence, the result is correct, but the paper’s proof needs revision; the model’s proof is correct and complete.