界面问题中由大的局部误差产生的二阶精度：离散格林函数分析
Second-Order Accuracy from Large Local Errors in Interface Problems: A Discrete Green's Function Analysis
- Bowling Green State University（鲍林格林州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究通过离散格林函数结构解释界面问题中局部大误差仍能达到二阶精度的现象，提出一类保守界面通量平衡格式，经一维、二维数值实验验证其有效性。
AI中文摘要：
有限差分法求解界面问题时，在界面附近常出现较大的局部截断误差，尤其当系数或解的导数存在不连续时更是如此。然而，许多此类格式仍能达到二阶精度，这一现象无法用标准的逐点相容性论证完全解释。本研究通过与基础保守差分算子相关的离散格林函数的结构，对该现象给出了精确解释。格林函数的显式表示显示其加权增量具有双平台结构。通过将数值误差用该格林核表示，我们证明占主导地位的界面截断误差虽各自为\boldsymbol{O(1)}阶，但具有抵消结构，可降低其对全局误差的有效贡献。精确的格林函数分析导出了一类保守界面通量平衡格式，其抵消机制在最大范数下可产生二阶精度。加权调和离散化作为该类格式的一个特例被包含，且其抵消特性得到直接验证。对于界面平行于坐标方向的笛卡尔网格，相同的保守抵消机制在穿过界面的网格线上依然存在。一维和二维数值实验直接验证了所预测的抵消行为及由此产生的二阶精度。这些结果表明，界面问题中的二阶精度可由局部截断误差与离散算子的全局结构之间的相互作用产生，而非仅源于逐点相容性。
英文摘要：
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.
