Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.13875

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

Zhixin Huo
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhixin Huo. 2026-08-14. PDE Realization and Structure-Aware Solvers for a Compact Two-Stage Fourth-Order IMEX Method. https://arxiv.org/abs/2608.13875

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗