Solving Bang-Bang Problems Using The Immersed Interface Method and Integer Programming

Both the paper and the candidate solution correctly target first-order consistency, stability, and convergence for the IIM-corrected one-step schemes on piecewise-smooth ODEs, and they agree on the finite-difference weights cn1=1/Δt, cn2=−1/Δt and on using jump data [T]=q and [Ṫ]=−K q + C[w] + [f] (state) and the analogous adjoint jumps. However, the paper’s printed correction term at an irregular state step uses (tn − α) in C̄n = (1/Δt)( q + [Ṫ](tn − α) ) (its eqs. (17)–(18)), which, when substituted into the discrete equation with RHS evaluated at tn, leaves a residual A+ − A− = [Ṫ] + O(Δt) rather than O(Δt). The proof in Proposition 1 then skips this cancellation and instead bounds terms as O((tn+1−α)^2)/Δt − O((tn−α)^2)/Δt without showing how the O(1) jump cancels, making the argument incomplete as written . The model reproduces the same correction and even derives that the left-hand side reduces to A+ + O(Δt), but then incorrectly asserts that the [Ṫ] difference cancels “via the explicit (tn − α)[Ṫ] term,” which has already been used earlier—leaving the same gap. Stability and convergence are invoked via standard one-step theorems with Lipschitz right-hand sides; the paper cites Hairer–Nørsett–Wanner and argues stability because corrections are bounded constants and then cites consistency+stability ⇒ convergence . These parts are directionally correct, but the irregular-step consistency calculation needs a corrected or clarified C̄n (very likely using (tn+1 − α) in the [Ṫ] term) to be complete. A minor inconsistency also appears in the introduction claiming “second order” while the rest of the paper and tables show first order .