Defect-Targeted Correction of BDF2 at Isolated Temporal Interfaces: A Local Analysis

The second-order backward differentiation formula (BDF2) is usually analyzed through smooth local expansions, which do not describe the first step crossing an isolated temporal interface. For piecewise-smooth solutions, the first post-interface residual is governed by a mixed one-sided crossing defect. This article derives a closed-form expansion for that defect and separates severe and benign regimes. In the severe regime, a first-derivative jump produces a generically order-one contribution, except at a special interior cancellation location. In the benign regime, where the first derivative is continuous but the second derivative is discontinuous, the leading jump-driven term is smaller but still exceeds the smooth-step residual scale. Interface-aware corrections cancel leading crossing terms at the residual level, whereas post-crossing restart leaves the first-crossing residual unchanged when the crossing value has already been produced by the BDF2 stencil. Direct residual tests, scalar and semidiscrete benchmarks, aligned and unaligned restart diagnostics, jump-data sensitivity tests, dimension-scaling checks, and a severe semidiscrete benchmark support this local interpretation. The formulation is BDF2-specific; higher-order BDF crossings require separate coefficient, history, and stability analyses. With accurate jump data, the correction is a local algebraic crossing-step modification that targets the mixed-side defect at the first post-interface update.