arXiv subjects

Shengtong Liang

Publications and source records attributed to Shengtong Liang.

3 recordsLinked to original sources

arXiv2026

An Unconditionally Linearly Convergent ADMM Approach for the Allen-Cahn Equation with Flory-Huggins Potential

The Allen-Cahn equation with Flory-Huggins potential is a fundamental and crucial model in phase field simulation for describing phase separation phenomena, which serves as a core tool in diverse branches of natural sciences. The numerical simulation of the Allen-Cahn equation is of great importance but poses significant challenges due to the strong nonlinearity and the presence of logarithmic singularities at $u=0,1$ in the Flory-Huggins potential. In this paper, we consider convex splitting schemes to %preserve this bound and guarantee unconditional unique solvability, which reduces the numerical simulation to solving a singular nonlinear system arising from spatial discretization at each time step. We propose an iterative solver that is specifically designed for such systems based on the alternating direction method of multipliers (ADMM) approach. The scheme possesses properties such as bound preserving and discrete energy stability. Building upon the recent unconditionally convergent ADMM framework for the Cahn-Hilliard equation (Li et al., 2026), our key theoretical contributions are twofold: (a) a proof of unconditional convergence when the multiplier update step size $α\in (0,\frac{\sqrt{5}+1}{2})$; (b) a rigorous establishment of the linear convergence for the embedded ADMM solver. This effectively liberates the solver from time-step constraints or strict separation conditions. Comprehensive numerical experiments validate our proposed ADMM framework, where its theoretical predictions are fully substantiated in practice, showcasing efficiency and robustness.

math.NA↗

arXiv2025

Overcoming logarithmic singularities in the Cahn-Hilliard equation with Flory-Huggins potential: An unconditionally convergent ADMM approach

The Cahn-Hilliard equation with Flory-Huggins potential serves as a fundamental phase field model for describing phase separation phenomena. Due to the presence of logarithmic singularities at $u=\pm 1$, the solution $u$ is constrained within the interval $(-1,1)$. While convex splitting schemes are commonly employed to preserve this bound and guarantee unconditional unique solvability, their practical implementation requires solving nonlinear systems containing singular logarithmic terms at each time step. This introduces significant challenges in both ensuring convergence of iterative solvers and maintaining the solution bounds throughout the iterations. Existing solvers often rely on restrictive conditions -- such as the strict separation property or small time step sizes -- to ensure convergence, which can limit their applicability. In this work, we introduce a novel iterative solver that is specifically designed for singular nonlinear systems, with the use of a variant of the alternating direction method of multipliers (ADMM). By developing a tailored variable splitting strategy within the ADMM framework, our method efficiently decouples the challenging logarithmic nonlinearity, enabling effective handling of singularities. Crucially, we rigorously prove the unconditional convergence of our ADMM-based solver, which removes the need for time step constraints or strict separation conditions. This allows us to fully leverage the unconditional solvability offered by convex splitting schemes. Comprehensive numerical experiments demonstrate the superior efficiency and robustness of our ADMM variant, strongly validating both our algorithmic design and theoretical results.

math.NA↗

arXiv2024

An asymptotic-preserving method for the three-temperature radiative transfer model

We present an asymptotic-preserving (AP) numerical method for solving the three-temperature radiative transfer model, which holds significant importance in inertial confinement fusion. A carefully designedsplitting method is developed that can provide a general framework of extending AP schemes for the gray radiative transport equation to the more complex three-temperature radiative transfer model. The proposed scheme captures two important limiting models: the three-temperature radiation diffusion equation (3TRDE) when opacity approaches infinity and the two-temperature limit when the ion-electron coupling coefficient goes to infinity. We have rigorously demonstrated the AP property and energy conservation characteristics of the proposed scheme and its efficiency has been validated through a series of benchmark tests in the numerical part.

math.NA↗