Residual-based a posteriori enclosures for peak differences in linear hyperbolic hydraulic models

Abstract Differences of numerical water-level maxima can be ambiguous when discretization error is comparable to the predicted contrast. This study investigates paired residual-based analytical enclosures in smooth linear periodic acoustic and advected shallow-water systems. Finite-volume/RK3 solutions are reconstructed continuously using Fourier and Hermite representations, with error estimates for individual fields and paired differences. Candidate-peak localization and an allowance for between-sample maxima yield sufficient analytical conditions for a sign decision. In the reported synthetic free-surface benchmark (N = 121), a computed contrast of -0.479 mm is unresolved by an independent allowance of 0.693 mm but resolved by a localized allowance of 0.205 mm. The fixed-station allowance is 0.123 mm versus an independent 0.692 mm. The method incurs estimator overhead, and no universal computational advantage over refinement is established. The results are confined to smooth linear periodic settings; floating-point evaluations are not interval-certified, and no nonlinear or field validation is claimed. Keywords: hyperbolic systems; finite volume; a posteriori error; peak localization; shallow-water model; paired residual