arXiv subjects

Zhixin Huo

Publications and source records attributed to Zhixin Huo.

7 recordsLinked to original sources

arXiv2026

PDE Realization and Structure-Aware Solvers for a Compact Two-Stage Fourth-Order IMEX Method

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.

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arXiv2026

An L-Stable Sequential Two-Stage Fourth-Order Method with ADER Trajectory Derivatives for Stiff Transport--Relaxation Systems

A fully implicit two-stage fourth-order two-derivative time discretization was introduced previously as a temporal method. This paper closes that sequential integrator for stiff transport--relaxation equations by pairing a conservative finite-volume residual $\mathcal L_h$ with its discrete trajectory derivative $\mathcal G_h^{\rm tr}=D\mathcal L_h\,\mathcal L_h$. An ADER/Cauchy--Kowalevski predictor provides interface states and physical time derivatives; differentiating the same numerical flux and taking shared face differences yields a conservative approximation $\widetilde{\mathcal G}_h$. For linear constant-coefficient balance laws, $\widetilde{\mathcal G}_h=\mathcal G_h^{\rm tr}=\mathcal L_h^2$ exactly, although the derivative operator is assembled independently rather than by squaring the residual matrix. For nonlinear discretizations, the fourth-order temporal theory applies to $\mathcal G_h^{\rm tr}$, while a trajectory-closure consistency estimate controls the ADER approximation. The two unknown stage vectors are solved successively through two $N$-unknown systems. The completed step is fourth order and L-stable; the parameter $C_q=5/183$ cancels the leading inverse-power term and changes the deep-stiff amplification from $O(|z|^{-1})$ to $O(|z|^{-2})$. For fixed compatible spatial spaces, a slow--fast decomposition proves a full-step asymptotic-preserving operator limit with an $O(δ)$ estimate and gives a preparation-dependent uniform-accuracy classification. Linear finite-volume, nonlinear relaxation, one- and two-dimensional damping, diffusion-limit, and modal experiments verify the corresponding closure, accuracy, stability, and singular-limit claims within their stated scopes.

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arXiv2026

Certified Seventh-Order Two-Derivative Hermite Deferred Correction via Node-Sweep Co-Design

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.

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arXiv2026

Arbitrary-Order Padé-Closed Anchored Two-Derivative Time Discretizations: $s$ Active Stages, Order $2s$, and $L$-Stability

An arbitrary-order family of implicit two-derivative one-step methods is constructed in an anchored active-stage formulation. At each information node the method uses both the vector field and its first total time derivative, enriching the local Hermite data without increasing the number of unknown stage states. With the known initial value retained as an anchor and $s$ unknown active stages, $2s$ Hermite moment conditions yield global order $2s$. The two remaining coefficients in each stage row are fixed by the second-subdiagonal Padé approximant $[s-1/s+1]_{e^z}$. For every ordered real node set, a Padé--Hermite basis theorem proves that the closure is unique, preserves all moment conditions, and gives $\det(I-zA-z^2\widehat A)=Q_s(z)$ and $R_s(z)=P_s(z)/Q_s(z)$. Hence the coupled stage system has no hidden poles and the accepted one-step map is $L$-stable (and therefore $A$-stable) for every positive integer $s$. Exact symbolic verification is reported through $s=6$, and high-precision computations confirm orders $2,4,6,$ and $8$ for the first four members. At equal active-stage count, comparisons with Gauss--Legendre and Radau IIA methods demonstrate the combined high-order accuracy and strong stiff damping of the construction over broad step-size ranges.

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arXiv2026

An L-Stable Implicit Two-Stage Fourth-Order Temporal Discretization Scheme for Lax-Wendroff-Type Solvers Applied to Stiff Problems

The explicit two-stage fourth-order (TSFO) temporal-spatial coupling method is efficient and compact but suffers severe time-step restrictions for stiff problems with multiple scales. To address Professor Jiequan Li's call for an implicit extension, this paper first constructs an implicit TSFO time discretization scheme using the method of undetermined coefficients and Taylor expansion. Second, using a model equation and the maximum modulus principle, sufficient conditions for L-stability are derived. Third, a Newton iteration accelerates convergence. Numerical experiments on classical stiff benchmarks show that the proposed implicit scheme achieves fourth-order temporal accuracy in two stages. Compared to the classical fourth-order implicit Runge-Kutta method, it allows larger stable time steps and reduces convergence errors by an order of magnitude. More importantly, this implicit scheme can be extended to construct an implicit TSFO temporal-spatial coupling method that captures flow-field correlations and handles strong discontinuities, fundamentally contrasting with method-of-lines approaches. Additionally, it unlocks Lax-Wendroff-type solvers to naturally and synchronously embed both stiff source terms and flow transport into time derivatives, thereby avoiding operator-splitting errors.

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arXiv2026

A Compact Two-Stage Fourth-Order Two-Derivative IMEX Method with Mixed Compatibility, High Implicit-Solve Efficiency,and Enhanced Stiff Decay

For additively split stiff evolution problems, classical fourth-order IMEX Runge-Kutta methods usually require several stages and complicated coupling order conditions. This paper proposes a compact fourth-order IMEX-type method based on a two-derivative formulation and analyzes its accuracy and stability properties. The proposed scheme incorporates the mixed explicit-implicit interaction directly through temporal derivatives evaluated along the full vector field, which ensures mixed compatibility for non-commuting split systems. With only one intermediate stage and two implicit solves per time step, the method achieves fourth-order accuracy while improving the accuracy obtained per implicit solve compared with classical multi-stage fourth-order IMEX-RK schemes. In addition, the method exhibits stronger damping of stiff modes in the purely implicit scalar limit and in the strong implicit-stiffness limit with a fixed explicit component. In the purely implicit scalar limit, its stability factor decays quadratically as the stiffness increases, whereas a representative classical fourth-order IMEX-RK reference method shows only linear decay. Numerical experiments on non-commuting split systems, scalar stiff-mode damping, increasing stiffness tests, and one- and two-dimensional advection-diffusion high-mode problems confirm the mixed consistency, the predicted stiff decay, and the smaller errors obtained under equal implicit-solve budgets in strongly stiff regimes.

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arXiv2026

A High-resolution Spatiotemporal Coupling Ghost Fluid Method for Two-Dimensional Compressible Multimedium Flows with Source Terms

While exact and approximate Riemann solvers are widely used, they exhibit two fundamental limitations: 1) Fail to represent continuous entropy transport processes, resulting in thermodynamic incompatibility that limits their applicability to compressible flows. 2) Consider only the effects of normal components at interfaces while neglecting the effects of tangential flux and source term, making them unsuitable for multidimensional problems and cases involving source terms. These limitations persist in Riemann problem-based ghost fluid methods. To address these challenges, we developed a novel spatiotemporal coupling high-resolution ghost fluid method featuring two key advancements: 1) Integration of nonlinear geometrical optics to properly account for thermodynamic entropy evolution. 2) Implementation of the Lax-Wendroff/Cauchy-Kowalevski approach to incorporate tangential fluxes and source term effects. These enhancements have been systematically applied to Riemann problem-based ghost fluid methods. Comprehensive numerical experiments demonstrate significant improvements in simulation accuracy and robustness compared to conventional approaches.

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