AI 中文总结
该研究构造满足标准假设的3×3矩阵微分方程，证明经典Strang投影分裂积分器存在鲁棒二阶收敛性的反例，其局部误差含非零h²项，无法升级为鲁棒二阶收敛。
AI 中文摘要
经典Strang投影分裂积分器被广泛观测到以二阶收敛，而当低秩近似中保留的最小奇异值趋于零时仍保持一致的误差分析仅能证明其一阶收敛。我们证明在标准假设下，这种差距是内在的。我们构造了3×3矩阵微分方程，其对时间为C²类、对矩阵变量光滑，具有秩为2的初始数据，满足一致有界性、Lipschitz、切向缺陷及正则性界。然而，精确子流Strang方法在由四个Strang步组成的单个周期驱动循环上的局部误差中存在非零的h²项。该循环的重复排除了全局二阶界，其常数和步长阈值与保留的奇异值无关。机制是与小奇异值相关的因子方向的快速旋转：一阶一致性得以保留，但不断变化的投影空间阻止了对称Strang组合通常预期的抵消。因此，仅在这些假设下，经典投影分裂方法的鲁棒一阶结果无法升级为鲁棒二阶结果。
英文摘要
The classical Strang projector-splitting integrator is widely observed to converge with order two, whereas the error analysis that remains uniform as the smallest singular value retained in the low-rank approximation tends to zero proves only order one. We show that this gap is intrinsic under the standard assumptions. We construct $3\times3$ matrix differential equations that are $C^2$ in time and smooth in the matrix variable, with rank-two initial data that satisfy uniform boundedness, Lipschitz, tangency-defect, and regularity bounds. Nevertheless, the exact-subflow Strang method has a nonzero $h^2$ term in the local error over one periodic forcing cycle consisting of four Strang steps. Repetition of that cycle rules out a global second-order bound whose constant and stepsize threshold are independent of the retained singular values. The mechanism is a rapid rotation of the factor directions associated with the small singular value: first-order consistency is preserved, but the changing projection spaces prevent the cancellation normally expected from a symmetric Strang composition. Hence the robust first-order result cannot, under these assumptions alone, be upgraded to robust second order for the classical projector-splitting method.
