arXiv subjects

Tiao Lu

Publications and source records attributed to Tiao Lu.

7 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 $\alpha \in (0,\frac{\sqrt{5}+1}{2})$; (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↗

arXiv2024

High Order Finite Difference Schemes for the Transparent Boundary Conditions and Their Applications in the 1D Schr\"odinger-Poisson Problem

The 1D Schr\"odinger equation closed with the transparent boundary conditions(TBCs) is known as a successful model for describing quantum effects, and is usually considered with a self-consistent Poisson equation in simulating quantum devices. We introduce discrete fourth order transparent boundary conditions(D4TBCs), which have been proven to be essentially non-oscillating when the potential vanishes, and to share the same accuracy order with the finite difference scheme used to discretize the 1D Schr\"odinger equation. Furthermore, a framework of analytic discretization of TBCs(aDTBCs) is proposed, which does not introduce any discretization error, thus is accurate. With the accurate discretizations, one is able to improve the accuracy of the discretization for the 1D Schr\"odinger problem to arbitrarily high levels. As numerical tools, two globally fourth order compact finite difference schemes are proposed for the 1D Schr\"odinger-Poisson problem, involving either of the D4TBCs or the aDTBCs, respectively, and the uniqueness of solutions of both discrete Schr\"odinger problems are rigorously proved. Numerical experiments, including simulations of a resistor and two nanoscale resonant tunneling diodes, verify the accuracy order of the discretization schemes and show potential of the numerical algorithm introduced for the 1D Schr\"odinger-Poisson problem in simulating various quantum devices.

math.NA↗

arXiv2016

Singularity-free Numerical Scheme for the Stationary Wigner Equation

For the stationary Wigner equation with inflow boundary conditions, its numerical convergence with respect to the velocity mesh size are deteriorated due to the singularity at velocity zero. In this paper, using the fact that the solution of the stationary Wigner equation is subject to an algebraic constraint, we prove that the Wigner equation can be written into a form with a bounded operator $\mathcal{B}[V]$, which is equivalent to the operator $\mathcal{A}[V]=\Theta[V]/v$ in the original Wigner equation under some conditions. Then the discrete operators discretizing $\mathcal{B}[V]$ are proved to be uniformly bounded with respect to the mesh size. Based on the therectical findings, a signularity-free numerical method is proposed. Numerical reuslts are proivded to show our improved numerical scheme performs much better in numerical convergence than the original scheme based on discretizing $\mathcal{A}[V]$.

math.NA↗

arXiv2016

Parity-decomposition and Moment Analysis for Stationary Wigner Equation with Inflow Boundary Conditions

We study the stationary Wigner equation on a bounded, one-dimensional spatial domain with inflow boundary conditions by using the parity decomposition in (Barletti and weifel, Trans. Theory Stat. Phys., 507--520, 2001). The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd $L^2$-space by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.

math.AP↗

arXiv2014

Convergence of Semi-discrete Stationary Wigner Equation with Inflow Boundary Conditions

Making use of the Whittaker-Shannon interpolation formula with shifted sampling points, we propose in this paper a well-posed semi-discretization of the stationary Wigner equation with inflow BCs. The convergence of the solutions of the discrete problem to the continuous problem is then analysed, providing certain regularity of the solution of the continuous problem.

math.NA↗

arXiv2013

Stationary Wigner Equation with Inflow Boundary Conditions: Will a Symmetric Potential Yield a Symmetric Solution?

Based on the well-posedness of the stationary Wigner equation with inflow boundary conditions given in (A. Arnold, H et al. J. Math. Phys., 41, 2000), we prove without any additional prerequisite conditions that the solution of the Wigner equation with symmetric potential and inflow boundary conditions will be symmetric. This improve the result in (D. Taj et al. Europhys. Lett., 74, 2006) which depends on the convergence of solution formulated in the Neumann series. By numerical studies, we present the convergence of the numerical solution to the symmetric profile for three different numerical schemes. This implies that the upwind schemes can also yield a symmetric numerical solution, on the contrary to the argument given in (D. Taj et al. Europhys. Lett., 74, 2006).

math-ph↗

arXiv2012

Quantum Hydrodynamic Model by Moment Closure of Wigner Equation

In this paper, we derive the quantum hydrodynamics models based on the moment closure of the Wigner equation. The moment expansion adopted is of the Grad type firstly proposed in \cite{Grad}. The Grad's moment method was originally developed for the Boltzmann equation. In \cite{Fan_new}, a regularization method for the Grad's moment system of the Boltzmann equation was proposed to achieve the globally hyperbolicity so that the local well-posedness of the moment system is attained. With the moment expansion of the Wigner function, the drift term in the Wigner equation has exactly the same moment representation as in the Boltzmann equation, thus the regularization in \cite{Fan_new} applies. The moment expansion of the nonlocal Wigner potential term in the Wigner equation is turned to be a linear source term, which can only induce very mild growth of the solution. As the result, the local well-posedness of the regularized moment system for the Wigner equation remains as for the Boltzmann equation.

math-ph↗