arXiv subjects

Maojun Li

Publications and source records attributed to Maojun Li.

12 recordsLinked to original sources

arXiv2026

Numerical solutions of an accurate diffuse interface model of the incompressible resistive MHD free surface flow

In this paper, we derive a new model to simulate the incompressible resistive magnetohydrodynamic (MHD) free surface flow. A thermodynamically consistent diffuse interface method is adopted to characterize the moving interface in the modeling process. The formal convergence of the proposed MHD free surface flow model to the sharp interface model is established via a matched asymptotic argument, and the model can be solved without the need for sophisticated free surface capturing schemes. We design a fully decoupled linear finite element scheme that preserves the divergence-free constraint of the magnetic field at a discrete level. The reliability and robustness of the proposed model and algorithm are validated through numerical investigations of the magnetic damping effect on bubble dynamics. In particular, we provide a quantitative numerical comparison of the present results with those obtained from an inductionless MHD model and a sharp interface arbitrary Lagrangian--Eulerian model.

math.NA↗

arXiv2026

A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density

In this paper, we present a numerical scheme designed for coupled systems of variable-topography shallow water flow and solute transport. By integrating a variable-density system with an expression for relative density of mixtures, a novel formulation of the coupled system is derived. To ensure the well-balanced property, auxiliary variables are introduced to reformulate the variable-density shallow water equations into a new form, which is then discretized using the discontinuous Galerkin (DG) method with the Lax-Friedrichs (LF) flux as the numerical flux. By selecting appropriate values for the auxiliary variables, we demonstrate that the proposed method accurately preserves steady-state solutions under still water conditions, thereby verifying its well-balanced nature. Furthermore, sufficient conditions for preserving the positivity of both water depth and concentration are proposed and rigorously proven. A positivity-preserving limiter is introduced to enforce these conditions. Finally, a series of numerical examples are conducted to validate the computational accuracy and effectiveness of the proposed method.

math.NA↗

arXiv2025

UniFucGrasp: Human-Hand-Inspired Unified Functional Grasp Annotation Strategy and Dataset for Diverse Dexterous Hands

Dexterous grasp datasets are vital for embodied intelligence, but mostly emphasize grasp stability, ignoring functional grasps needed for tasks like opening bottle caps or holding cup handles. Most rely on bulky, costly, and hard-to-control high-DOF Shadow Hands. Inspired by the human hand's underactuated mechanism, we establish UniFucGrasp, a universal functional grasp annotation strategy and dataset for multiple dexterous hand types. Based on biomimicry, it maps natural human motions to diverse hand structures and uses geometry-based force closure to ensure functional, stable, human-like grasps. This method supports low-cost, efficient collection of diverse, high-quality functional grasps. Finally, we establish the first multi-hand functional grasp dataset and provide a synthesis model to validate its effectiveness. Experiments on the UFG dataset, IsaacSim, and complex robotic tasks show that our method improves functional manipulation accuracy and grasp stability, demonstrates improved adaptability across multiple robotic hands, helping to alleviate annotation cost and generalization challenges in dexterous grasping. The project page is at https://haochen611.github.io/UFG.

cs.RO↗

arXiv2025

On the direct and inverse electromagnetic scattering in a parallel-plate waveguide

This paper devotes to providing rigorous theoretical analysis of the wellposedness of the direct problem and the uniqueness of the inverse problem of electromagnetic scattering in a parallel-plate waveguide. The direct problem is reduced to an equivalent boundary value problem on a bounded domain by introducing an exact transparent boundary condition in terms of the electric-to-magnetic Calder\'on operator which can be explicitly represented as a series expansion. Then the wellopsedness of the reduced problem in appropriate Sobolev spaces is proved via the variational approach provided by some necessary properties of the Calder\'on operator and Helmholtz decomposition. Relying on the Green's representation formula and a reciprocity relation, the probe method, finally, is utilized to show the uniqueness of the inverse obstacle problem.

math.AP↗

arXiv2025

Modeling and simulation of inductionless magnetohydrodynamic free surface problems with unmatched densities

We propose a new diffuse interface model for simulating an inductionless magnetohydrodynamic (MHD) free surface problem. By using the Onsager's variational principle and the laws of thermodynamics, we derive a thermodynamically consistent system that couples the Cahn--Hilliard equation modeling phase separation, the Navier--Stokes equations governing fluid motion, and a generalized Darcy's law accounting for electromagnetic effects. In contrast to existing diffuse interface MHD models, the proposed model can handle general material properties in practical engineering applications. Furthermore, through asymptotic arguments, we investigate the sharp interface limit, and then demonstrate that the classical sharp interface model can be recovered as the interface thickness approaches zero, theoretically validating the proposed diffuse interface model as an approximate approach. An efficient decoupled, linear, and charge-conservative finite element scheme is designed, and it significantly facilitates the large-scale and accurate numerical simulations involving large parameter ratios. Finally, we present several three-dimensional numerical experiments of magnetic damping effects on bubble dynamics for the demonstration of the capability of the proposed model and method in capturing complex MHD phenomena.

math.NA↗

arXiv2025

Optimal error estimates of a second-order temporally finite element method for electrohydrodynamic equations

In this work, we mainly present the optimal convergence rates of the temporally second-order finite element scheme for solving the electrohydrodynamic equation. Suffering from the highly coupled nonlinearity, the convergence analysis of the numerical schemes for such a system is rather rare, not to mention the optimal error estimates for the high-order temporally scheme. To this end, we abandon the traditional error analysis method following the process of energy estimate, which may lead to the loss of accuracy. Instead, we note that the charge density also possesses the "energy" decaying property directly derived by its governing equation, although it does not appear in the energy stability analysis. This fact allows us to control the error terms of the charge density more conveniently, which finally leads to the optimal convergence rates. Several numerical examples are provided to demonstrate the theoretical results, including the energy stability, mass conservation, and convergence rates.

math.NA↗

arXiv2025

Mass conservation, positivity and energy identical-relation preserving scheme for the Navier-Stokes equations with variable density

In this paper, we consider a mass conservation, positivity and energy identical-relation preserving scheme for the Navier-Stokes equations with variable density. Utilizing the square transformation, we first ensure the positivity of the numerical fluid density, which is form-invariant and regardless of the discrete scheme. Then, by proposing a new recovery technique to eliminate the numerical dissipation of the energy and to balance the loss of the mass when approximating the reformation form, we preserve the original energy identical-relation and mass conservation of the proposed scheme. To the best of our knowledge, this is the first work that can preserve the original energy identical-relation for the Navier-Stokes equations with variable density. Moreover, the error estimates of the considered scheme are derived. Finally, we show some numerical examples to verify the correctness and efficiency.

math.NA↗

arXiv2024

Efficient finite element schemes for a phase field model of two-phase incompressible flows with different densities

In this paper, we present two multiple scalar auxiliary variable (MSAV)-based, finite element numerical schemes for the Abels-Garcke-Gr{\"u}n (AGG) model, which is a thermodynamically consistent phase field model of two-phase incompressible flows with different densities. Both schemes are decoupled, linear, second-order in time, and the numerical implementation turns out to be straightforward. The first scheme solves the Navier-Stokes equations in a saddle point formulation, while the second one employs the artificial compressibility method, leading to a fully decoupled structure with a time-independent pressure update equation. In terms of computational cost, only a sequence of independent elliptic or saddle point systems needs to be solved at each time step. At a theoretical level, the unique solvability and unconditional energy stability (with respect to a modified energy functional) of the proposed schemes are established. In addition, comprehensive numerical simulations are performed to verify the effectiveness and robustness of the proposed schemes.

math.NA↗

arXiv2023

On a Robin-type non-singular coupling scheme for solving the wave scattering problems

This paper studies a non-singular coupling scheme for solving the acoustic and elastic wave scattering problems and its extension to the problems of Laplace and Lam\'e equations and the problem with a compactly supported inhomogeneity is also briefly discussed. Relying on the solution representation of the wave scattering problem, a Robin-type artificial boundary condition in terms of layer potentials whose kernels are non-singular, is introduced to obtain a reduced problem on a bounded domain. The wellposedness of the reduced problems and the a priori error estimates of the corresponding finite element discretization are proved. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

math.NA↗

arXiv2022

An efficient and energy decaying discontinuous Galerkin method for Maxwell's equations for the Cole-Cole dispersive medium

In this work, we investigate the propagation of electromagnetic waves in the Cole-Cole dispersive medium by using the discontinuous Galerkin (DG) method to solve the coupled time-domain Maxwell's equations and polarization equation. We define a new and sharpened total energy function for the Cole-Cole model, which better describes the behaviors of the energy than what is available in the current literature. A major theme in the time-domain numerical modeling of this problem has been tackling the difficulty of handling the nonlocal term involved in the time-domain polarization equation. Based on the diffusive representation and the quadrature formula, we derive an approximate system, where the convolution kernel is replaced by a finite number of auxiliary variables that satisfy local-in-time ordinary differential equations. To ensure the resulted approximate system is stable, a nonlinear constrained optimization numerical scheme is established to determine the quadrature coefficients. By a special choice of the numerical fluxes and projections, we obtain {for the constant coefficient case } an optimal-order convergence result for the semi-discrete DG scheme. The temporal discretization is achieved by the standard two-step backward difference formula and a fast algorithm with linear complexity is constructed. Numerical examples are provided for demonstrating the efficiency of the proposed algorithm, validating the theoretical results and illustrating the behaviors of the energy.

math.NA↗

arXiv2019

Numerical simulation of a coupled system of Maxwell equations and a gas dynamic model

It is known that both linear and nonlinear optical phenomena can be produced when the plasmon in metallic nanostructures are excited by the external electromagnetic waves. In this work, a coupled system of Maxwell equations and a gas dynamic model including a quantum pressure term is employed to simulate the plasmon dynamics of free electron fluid in different metallic nanostructures using a discontinuous Galerkin method in two dimensions. Numerical benchmarks demonstrate that the proposed numerical method can simulate both the high order harmonic generation and the nonlocal effect from metallic nanostructures. Based on the switch-on-and-off investigation, we can conclude that the quantum pressure term in gas dynamics is responsible for the bulk plasmon resonance. In addition, for the dielectric-filled nano-cavity, a coupled effective polarization model is further adopted to investigate the optical behavior of bound electrons. Concerning the numerical setting in this work, a strengthened influence of bound electrons on the generation of high order harmonic waves has been observed.

physics.comp-ph↗

arXiv2019

A CDG-FE method for the two-dimensional Green-Naghdi model with the enhanced dispersive property

In this work, we investigate numerical solutions of the two-dimensional shallow water wave using a fully nonlinear Green-Naghdi model with an improved dispersive effect. For the purpose of numerics, the Green-Naghdi model is rewritten into a formulation coupling a pseudo-conservative system and a set of pseudo-elliptic equations. Since the pseudo-conservative system is no longer hyperbolic and its Riemann problem can only be approximately solved, we consider the utilization of the central discontinuous Galerkin method which possesses an important feature of needlessness of Riemann solvers. Meanwhile, the stationary elliptic part will be solved using the finite element method. Both the well-balanced and the positivity-preserving features which are highly desirable in the simulation of the shallow water wave will be embedded into the proposed numerical scheme. The accuracy and efficiency of the numerical model and method will be illustrated through numerical tests.

math.NA↗