Second-Order Accuracy from Large Local Errors in Interface Problems: A Discrete Green's Function Analysis
Finite difference methods for interface problems often exhibit large local truncation errors near the interface, particularly when discontinuities in coefficients or solution derivatives are present. Nevertheless, many such schemes achieve second-order accuracy, a phenomenon not fully explained by standard pointwise consistency arguments. In this work, we provide a precise explanation of this behavior through the structure of the discrete Green's function associated with the underlying conservative difference operator. An explicit representation of the Green's function reveals a two-plateau structure in its weighted increments. By expressing the numerical error in terms of this Green kernel, we show that the dominant interface truncation errors, although individually of order \(O(1)\), possess a cancellation structure that reduces their effective contribution to the global error. The exact Green's-function analysis leads to a class of conservative interface flux--balance schemes for which the cancellation mechanism yields second-order accuracy in the maximum norm. The weighted harmonic discretization is included as a particular member, and its cancellation property is established directly. For Cartesian grids with a flat interface parallel to a coordinate direction, the same conservative cancellation mechanism persists along grid lines crossing the interface. Numerical experiments in one and two dimensions directly illustrate the predicted cancellation behavior and the resulting second-order accuracy. These results demonstrate that second-order accuracy in interface problems can arise from the interaction between localized truncation errors and the global structure of the discrete operator, rather than from pointwise consistency alone.