2026/07/31 by Zhixin Huo
Mathematics · Computer Science · #math.NA #cs.NA
arxiv created 2026/07/31 · arxiv updated 2026/08/03
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 Jstiff<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.