刚性分裂问题的紧凑两阶段四阶IMEX方法的PDE实现与结构感知求解器
PDE Realization and Structure-Aware Solvers for a Compact Two-Stage Fourth-Order IMEX Method
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中文总结 AI 辅助
本文针对刚性分裂问题，开发了紧凑两阶段四阶IMEX方法的PDE实现与结构感知求解器，通过ADER局部演化、非对称残差容差等技术，在保证四阶精度的同时提升效率，数值实验验证了其性能。
中文摘要 AI 辅助
本文针对刚性分裂问题，开发了一种用于紧凑两阶段四阶双导数隐式-显式时间离散化的PDE实现与求解器框架。核心难点在于一致性与成本的耦合：需要全场时间微分以保持时间积分器的混合显式-隐式相互作用，但该相互作用会拓宽隐式阶段算子，可能使每次求解的成本大幅增加。本文证明二阶ADER/Cauchy-Kowalevski局部演化足以支撑四阶外层组合，建立了光滑重构一致性与完全离散误差平衡，并推导了分裂场自微分产生的主导李括号缺陷。不精确阶段分析给出非对称中点与端点残差容差，可保持四阶精度。对于拓宽的阶段，以无矩阵方式保留完整混合作用，仅在逆运算中近似主导刚性物理，由此得到二次与移位预条件子、傅里叶与多级实现、半线性反应-扩散约化以及松弛系统的源局部消元。精确二次抵消、扩散主导的傅里叶估计以及Jin-Xin松弛的ε/Δx界解释了主要求解器机制。数值消融实验验证了一致性与容差结果，而在两个网格上开展的二维Brusselator研究显示，在有用精度范围内，其误差与墙钟时间的表现更优。经典刚性前沿基准还识别出单独的空间激波-源限制。研究结果支持与区域相关的效率主张，而非通用加速。
英文摘要
This paper develops a PDE realization and solver framework for a compact two-stage fourth-order two-derivative implicit--explicit time discretization for stiff split problems. The key difficulty is a consistency--cost coupling: full-field temporal differentiation is required to preserve the mixed explicit--implicit interactions of the time integrator, but the same interactions widen the implicit stage operators and can make each solve substantially more expensive. We show that a second-order ADER/Cauchy--Kowalevski local evolution is sufficient for the fourth-order outer composition, establish smooth reconstruction consistency and a fully discrete error balance, and derive the leading Lie-bracket defect produced by self differentiation of the split fields. An inexact-stage analysis gives asymmetric midpoint and endpoint residual tolerances that preserve fourth-order accuracy. For the widened stages, the complete mixed action is retained matrix-free while only dominant stiff physics is approximated in the inverse. This yields quadratic and shifted preconditioners, Fourier and multilevel realizations, a semilinear reaction--diffusion reduction, and source-local elimination for relaxation systems. Exact quadratic cancellation, a diffusion-dominated Fourier estimate, and an $\varepsilon/Δx$ bound for Jin--Xin relaxation explain the main solver mechanisms. Numerical ablations verify the consistency and tolerance results, while a two-dimensional Brusselator study on two grids shows favorable error-versus-wall-time behavior over a useful accuracy range. A classical stiff-front benchmark also identifies the separate spatial shock--source limitation. The results support a regime-dependent efficiency claim rather than a universal speedup.
