慢-快响应修正用于紧凑四阶IMEX二导数方法

中文摘要 AI 辅助

一种紧凑的两阶段四阶二导数IMEX方法在松弛时间与时间步长相当的情况下，可能保持其经典的非交换四阶展开和L稳定的刚性阻尼，同时失去高阶精度。我们询问是否可以在不重新设计原始两阶段、两次隐式求解核心的情况下恢复均匀的四阶精度。对于具有分离慢谱子空间的线性松弛系统，我们引入了一种慢-快响应修正（SFRC）：精确的慢投影由不变的紧凑IMEX步进推进并投影回慢子空间，而快补集由其精确半群推进。对于固定的非零松弛时间，修正为$O(\dt^5)$，因此通过四阶的基本混合展开不变，纯快响应保持L稳定。一个抽象的慢-快论证将全状态误差归结为慢块缺陷，并将均匀的$O(\dt^5)$局部估计转化为均匀的$O(\dt^4)$全局估计。对于每个Jin-Xin傅里叶模式，所需的局部估计通过精确的投影放大因子分解和$x=\eps/\dt$中的均匀界来证明。特别地，在限制到精确有限$\eps$慢特征空间后，不变的紧凑核心已经具有均匀的五阶单步缺陷。一个二维三变量模型在固定傅里叶网格上产生相同的机制。符号验证以及一维和二维实验证实了分析预测。因此，SFRC是一个严格的线性基准，识别出足以调和紧凑性、混合四阶精度、刚性阻尼和均匀四阶时间精度的有限$\eps$慢-快信息。

英文摘要

A compact two-stage fourth-order two-derivative IMEX method may retain its classical non-commuting fourth-order expansion and L-stable stiff damping while losing high-order accuracy when the relaxation time is comparable with the time step. We ask whether uniform fourth-order accuracy can be recovered without redesigning the original two-stage, two-implicit-solve core. For linear relaxation systems with a separated slow spectral subspace, we introduce a slow--fast response correction (SFRC): the exact slow projection is advanced by the unchanged compact IMEX step and projected back to the slow subspace, while the fast complement is advanced by its exact semigroup. For fixed nonzero relaxation time, the correction is $O(\dt^5)$, so the base mixed expansion through degree four is unchanged, and the pure-fast response remains L-stable. An abstract slow--fast argument reduces the full-state error to a slow-block defect and converts a uniform $O(\dt^5)$ local estimate into a uniform $O(\dt^4)$ global estimate. For each Jin--Xin Fourier mode, the required local estimate is proved by an exact projected-amplification factorization and a uniform bound in $x=\eps/\dt$. In particular, the unchanged compact core already has a uniform fifth-order one-step defect after restriction to the exact finite-$\eps$ slow eigenspace. A two-dimensional three-variable model yields the same mechanism on fixed Fourier grids. Symbolic verification and one- and two-dimensional experiments corroborate the analytical predictions. SFRC is therefore a rigorous linear benchmark identifying the finite-$\eps$ slow--fast information sufficient to reconcile compactness, mixed fourth-order accuracy, stiff damping, and uniform fourth-order time accuracy.

发表机构

• School of Mathematics and Information Science, Henan Polytechnic University（河南理工大学数学与信息科学学院）

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