AI 中文总结
本文针对周期标量平流的一阶迎风格式有限体积法，推导有限视界观测秩定律，提出锚定通量散度队列方法，区分精确可观测性与稳定可恢复性，证明非恒定分量在有界扰动下的收缩性，为相关模型提供可求解基准。
AI 中文摘要
粗有限体积平均值通常无法构成可预测状态：实际界面通量虽可完成保守更新，但具有相同父平均值的不同细网格状态会产生不同的未来粗网格演化历史。本文针对采用前向欧拉时间积分的一阶迎风格式有限体积法离散的周期标量平流问题，分析了上述失效现象。推导得到有限视界观测秩定律：每增加一次观测，会暴露一层新的子单元，且贡献的独立方向数比父单元数少1，直至未解析层耗尽。集中式预测因此要求每暴露一层，额外坐标数比父单元数少1；而乘积局部预测则要求每个父单元对应一层一个坐标。锚定通量散度队列可达到集中式预测的界，识别周期通量 gauge，且在饱和时与父平均值共同构成最小自主可预测状态。随后区分了精确可观测性与稳定可恢复性：当Courant数减小时， collar-to-queue映射会迅速出现病态，导致代数可见的延迟信息可能低于规定数值容差。对于严格介于0和1之间的Courant数，本文证明在有界算术扰动下，非恒定分量会发生收缩，同时对平均漂移、显式扰动邻域及网格相关衰减时间进行了单独控制。数值实验展示了秩阶梯、有效秩损失、队列条件数、阶跃函数扁平化、延迟粗网格分离及后续耗散衰减过程。上述结果为评估粗网格、降阶、多尺度及学习型科学模型的状态充分性提供了可求解的基准。
英文摘要
Coarse finite-volume averages do not generally form a predictive state: realized interface fluxes close a conservative update, but distinct fine-grid states with identical parent averages can generate different future coarse histories. We analyze this failure for periodic scalar advection discretized by a first-order upwind finite-volume method with forward Euler time integration. We derive a finite-horizon observation-rank law: each additional observation exposes one new child-cell layer and contributes one fewer independent direction than the number of parent cells, until the unresolved layers are exhausted. Centralized prediction therefore requires one fewer additional coordinate than the number of parent cells per exposed layer, whereas product-local prediction requires one coordinate per layer in each parent. An anchored flux-divergence queue attains the centralized bound, identifies the periodic flux gauge, and, at saturation, forms a minimal autonomous predictive state with the parent averages. We then distinguish exact observability from stable recoverability. The collar-to-queue map becomes rapidly ill-conditioned as the Courant number decreases, so algebraically visible delayed information may fall below a prescribed numerical tolerance. For Courant numbers strictly between zero and one, we prove contraction of the nonconstant component under bounded arithmetic perturbations, with separate control of mean drift, an explicit perturbation neighborhood, and a grid-dependent decay time. Numerical experiments illustrate the rank ladder, effective-rank loss, queue conditioning, step-function flattening, and delayed coarse separation followed by dissipative decay. The results provide a solvable benchmark for assessing state sufficiency in coarse, reduced, multiscale, and learned scientific models.
Comments27 pages. Code, data, and executable certificates: https://github.com/UniversityOfNicosia/certified-simulation (release finite-horizon-memory-v1, DOI 10.5281/zenodo.21859861)
