arXiv subjects

Huini Liu

Publications and source records attributed to Huini Liu.

3 recordsLinked to original sources

arXiv2026

Global well-posedness of the linearized R13 moment equations with Onsager boundary conditions

This paper establishes the global well-posedness of the linearized regularized 13-moment (R13) equations for rarefied gas flows. We first derive an entropy inequality for the system on bounded domains subject to Onsager boundary conditions. For the steady-state problem, well-posedness is proved via the Ladyzhenskaya-Babuska-Brezzi (LBB) theorem, facilitated by novel boundary-related Korn-type inequalities. Furthermore, leveraging the Lumer-Phillips theorem, we extend these results to guarantee the global well-posedness of the time-dependent R13 equations. Our theoretical framework uniformly accommodates the models for both Maxwell and general non-Maxwell molecules.

math.AP↗

arXiv2026

A fully decoupled and structure-preserving relaxation Crank--Nicolson finite element method for Gross--Pitaevskii--Poisson model

We propose a fully decoupled, structure-preserving relaxation Crank--Nicolson finite element method (FEM) for the coupled Gross--Pitaevskii--Poisson (GPP) system modeling ultracold plasmas. By introducing suitable auxiliary variables to reformulate the nonlinear interaction and charge density terms, the original system is recast into an equivalent form that enables a linear, fully decoupled numerical scheme. The proposed method preserves key physical invariants, including the mass of each component and a modified discrete energy, at the fully discrete level. We establish the well-posedness and uniqueness of the scheme and rigorously derive optimal error estimates, achieving second-order accuracy in time and optimal $(k+1)$-th order convergence in space for $P^k$ finite element approximations. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the method in preserving conservation properties and accurately capturing complex dynamical behaviors of the coupled GPP system.

math.NA↗

arXiv2024

A structure-preserving relaxation Crank-Nicolson finite element method for the Schr\"{o}dinger-Poisson equation

In this paper, we propose a mass- and modified energy-conservative relaxation Crank-Nicolson finite element method for the Schr\"{o}dinger-Poisson equation. Utilizing only a single auxiliary variable, we simultaneously reformulate the distinct nonlinear terms present in both the Schr\"{o}dinger equation and the Poisson equation into their equivalent expressions, constructing a system equivalent to the original Schr\"{o}dinger-Poisson equation. Our proposed scheme, derived from this equivalent system, is implemented linearly, avoiding the need for iterative techniques to solve the nonlinear equation. Additionally, it is executed sequentially, eliminating the need to solve a coupled large linear system. We in turn rigorously derive the optimal error estimates for the proposed scheme, demonstrating second order accuracy in time and $(k+1)$th order accuracy in space when employing polynomials of degree up to $k$. Numerical experiments validate the accuracy and effectiveness of our method and emphasize its conservation properties over long-time simulations.

math.NA↗