基于节点-扫描协同设计的可证七阶双导数Hermite延迟校正方法
Certified Seventh-Order Two-Derivative Hermite Deferred Correction via Node-Sweep Co-Design
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中文总结 AI 辅助
本文针对扩散主导半线性问题，通过节点-扫描协同设计提出可证七阶双导数Hermite延迟校正方法，设计的Accuracy-P40方法可降低计算量并满足刚性条件，经测试验证了阶数与性能。
中文摘要 AI 辅助
双导数Hermite延迟校正结合了高配置阶数与顺序单状态求解，但该方法的收敛性同时依赖于节点与校正扫描。本文针对扩散主导的半线性问题，对上述要素进行协同设计：在三个子区间、H4预测子及两次校正的设置下，完整的七阶B级数缺陷在48维根树空间中秩为1，两次校正对单向五阶预测子缺陷应用两个一元嫁接，因此单个标量链系数控制所有非线性主误差条件；有理节点与孤立代数校正参数可抵消该系数，且与八阶链多项式互素，证明其经典阶数恰好为七。互补设计Accuracy-P40保持通用六阶，同时将完整主误差范数降至LGL--L3值的9.8%，且满足J_stiff<0.40。本文还区分了重复校正的收敛性与固定扫描次数后的绝对稳定性：两次校正具有有限负实稳定性区间，而第三次校正为新设计恢复了远刚性输出阻尼；高精度非线性阶数测试验证了六阶与七阶，Allen--Cahn和Cahn--Hilliard计算表明Accuracy-P40减少了校正与Krylov计算量。
英文摘要
Two-derivative Hermite deferred correction combines high collocation order with sequential single-state solves, but the stopped method depends jointly on the nodes and the correction sweep. We co-design these ingredients for diffusion-dominated semilinear problems. For three subintervals, an H4 predictor, and two corrections, the complete order-seven B-series defect has rank one in the 48-dimensional rooted-tree space: two corrections apply two unary graftings to the one-directional order-five predictor defect. Hence one scalar chain coefficient controls every nonlinear principal-error condition. Rational nodes and an isolated algebraic correction parameter cancel this coefficient; coprimality with the order-eight chain polynomial proves classical order exactly seven. A complementary design, Accuracy-P40, retains generic sixth order but reduces the complete principal-error norm to $9.8\%$ of the LGL--L3 value while satisfying $J_{\mathrm{stiff}}<0.40$. We also distinguish convergence of repeated corrections from absolute stability after a fixed number of sweeps: two corrections have finite negative-real stability intervals, whereas a third correction restores far-stiff output damping for the new designs. High-precision nonlinear order tests verify sixth versus seventh order, and Allen--Cahn and Cahn--Hilliard calculations show that Accuracy-P40 reduces correction and Krylov work.
