arXiv subjects

Xiaoming Zheng

Publications and source records attributed to Xiaoming Zheng.

10 recordsLinked to original sources

arXiv2026

A spectral-vanishing-viscosity stabilization of a higher-order consistent splitting scheme for the Navier-Stokes equations

Huang and Shen developed a novel class of high-order BDF-IMEX consistent-splitting schemes for the incompressible Navier-Stokes equations, giving the first rigorous stability and error analysis for a fully decoupled splitting scheme of temporal order higher than two. Extending their analysis from unit viscosity to arbitrary viscosity, this work reveals that the error upper bound coefficient contains inverse powers of the viscosity. Our numerical experiments show that the scheme can break down at high Reynolds number. To save the scheme from this failure, we stabilize it by adding to the velocity update a symmetric positive-semidefinite spectral vanishing viscosity operator, built from the directionally applied Maday-Kaber-Tadmor kernel, which selectively damps the high, under-resolved modes at no additional asymptotic cost and leaves the structure of the error analysis intact. We establish stability and error estimates for the stabilized scheme in which the spectral vanishing viscosity provides viscosity-independent coercive control of the high modes. Three two-dimensional tests demonstrate the robustness and accuracy of the stabilized scheme. For a manufactured solution, the stabilized scheme retains its design order for k=2,3,4, whereas the unstabilized scheme diverges. For a perturbed Kovasznay flow, it accurately resolves the boundary layer at Re=10^4 and drives the perturbation back to the steady state, while the unstabilized scheme blows up. For the Kelvin-Helmholtz instability problem, it reproduces the reference integral diagnostics throughout the reliable regime, whereas the unstabilized scheme produces spurious solutions or blows up.

math.NA↗

arXiv2026

A consistent-splitting generalized scalar auxiliary variable scheme for the perturbed Boussinesq system

We propose and analyze a second-order consistent-splitting scheme, based on the generalized scalar auxiliary variable (GSAV) approach, for the two-dimensional perturbed Boussinesq system. The system is obtained by subtracting a stable, linearly stratified hydrostatic equilibrium from the standard Boussinesq system. The time discretization extends the consistent-splitting generalized BDF2 framework of Huang and Shen [17] for the Navier-Stokes equations, treating the nonlinear convection and advection together with the linear buoyancy and stratification couplings explicitly, so that each time step reduces to a small number of decoupled linear systems. We prove an unconditional weak stability theorem for the GSAV scheme and derive optimal second-order error estimates for the velocity, pressure, and temperature. A careful tracing reveals that the error constant depends on the inverse viscosity and inverse thermal diffusivity through a quadruply-nested exponential, so the scheme is not robust as either tends to zero. Numerical experiments confirm the second-order convergence and reproduce the expected internal-wave dynamics and exponential relaxation toward hydrostatic balance in a long-time stratified-flow simulation.

math.NA↗

arXiv2026

Viscosity in error upper bound for a consistent splitting scheme of the Navier-Stokes equations

This paper investigates the role of viscosity in the error upper bounds of a consistent splitting scheme for the Navier-Stokes equations proposed by Huang and Shen [5]. In their original analysis the viscosity is fixed to unity. By following and extending their proof methodology while keeping the viscosity symbolic, we obtain an H1 velocity error bound that contains negative powers of viscosity, indicating that the scheme is not robust as viscosity tends zero. To establish this bound we refine a theorem in [8] on the constant in the Stokes pressure estimate, which is crucial to the error analysis. A targeted numerical experiment based on a perturbation of the Kovasznay flow corroborates this analytical prediction: the scheme of [5] blows up at high Reynolds number, and a comparison with a fully implicit Newton solver and with the time-dependent Stokes counterpart of the same scheme localizes the failure to the explicit treatment of the convection term.

math.NA↗

arXiv2025

Unveiling Explicit Patterns: Exact Steady States and Stability in a Confined Chemotaxis Model

Inspired by Carrillo-Li-Wang's work [Proc. London Math. Soc., 2021] on stationary solutions to the singular Keller-Segel system, this paper presents a novel family of explicit steady-state solutions for the same model on a bounded interval, expressed in terms of trigonometric and hyperbolic functions. Under Dirichlet boundary conditions and within a biologically stable parameter regime, these solutions, including singular types such as secant and cosecant, are rigorously derived and analyzed. Their stability is established via energy methods, yielding precise thresholds for pattern persistence. These results provide valuable benchmarks for numerical validation and offer insights into boundary-driven pattern formation.

math.AP↗

arXiv2025

Iterative projection method for unsteady Navier-Stokes equations with high Reynolds numbers

A new iterative projection method is proposed to solve the unsteady Navier-Stokes equations with high Reynolds numbers. The convectional projection method attempts to project the intermediate velocity to the divergence free space only once per time step. However, such a velocity is not genuinely divergence free in general practice, which can yield large errors when the Reynolds number is high. The new method has several important features: the BDF2 time discretization, the skew-symmetric convection in a semi-implicit form, two modulating parameters, and the iterative projections in each time step. A major difficulty in the proof of iteration convergence is the nonlinear convection. We solve this problem by first analyzing the non-convective scheme with a focus on the spectral properties of the iterative matrix, and then employing a delicate perturbation analysis for the convective scheme. The work achieves the weakly divergence free velocity (strongly divergence free for divergence free finite element spaces), and the rigorous stability and error analysis when the iterations converge. The three dimensional numerical tests confirm that this new method can effectively treat high Reynolds numbers with only a few iterations per time step, where the convectional projection method and the iterative projection method with the explicit convection would fail.

math.NA↗

arXiv2024

Boundary control for optimal mixing via Stokes flows and numerical implementation

This work develops scientific computing techniques to further the exploration of using boundary control alone to optimize mixing in Stokes flows. The theoretical foundation including mathematical model and the optimality conditions for solving the optimal control has been established by Hu and Wu in a series of work. The scalar being mixed is purely advected by the flow and the control is exerted tangentially on the domain boundary through the Navier slip conditions. The control design is motivated by the physical observations that the moving or rotating walls accelerate mixing. A gradient descent-based optimization algorithm is designed. A critical problem is the computation of the Gateaux derivative or the gradient of the cost functional. Two methods are proposed: one is based on the Variational Formula (VF) and one utilizes Algorithmic Differentiation (AD). The convergence of the algorithm is studied and various designs of boundary control using cosine and sine functions with time segmentation are computed. The algorithm has a first order convergence rate and the VF method is more efficient by taking only one third of the time as the AD method when the dimension of control basis is large. The numerical implementations show that the boundary control produces similar mixing results as internal mixings in the existing literature. The mixing effect becomes better when more diverse basis control functions and more time segmentation are utilized. It is shown that the mixing decay rate in time follows power rules, approximately. The numerical study in this work suggests that boundary control alone could be an effective strategy for mixing in incompressible fluid flows.

math.OC↗

arXiv2023

Numerical algorithms and simulations of boundary dynamic control for optimal mixing in unsteady Stokes flows

This work develops an efficient and accurate optimization algorithm to study the optimal mixing problem driven by boundary control of unsteady Stokes flows, based on the theoretical foundation laid by Hu and Wu in a series of work. The scalar being mixed is purely advected by the flow and the control is a force exerted tangentially on the domain boundary through the Navier slip conditions. The control design has potential applications in many industrial processes such as rotating wall driven mixing, micromixers with acoustic waves, and artificial cilia mixing. The numerical algorithms have high complexity, high accuracy demand, and high computing expense, due to the multiscale nature of the mixing problem and the optimization requirements. A crucial problem is the computation of the G$â$teaux derivative of the cost functional. To this end, a hybrid approach based on variational formula and finite difference is built with high accuracy and efficiency to treat various types of control input functions. We have experimented with various optimization schemes including the steepest descent algorithm, the conjugate gradient method and two line search options (backtracking and exact line search). We are able to identify and implement the best combinations. The numerical simulations show that the mixing efficacy is limited when only one single type of control is applied, but can be enhanced when more diverse control types and more time segmentation are utilized. The mix-norm in the optimal mixings decays exponentially. The numerical study in this work demonstrates that boundary control alone could be an effective strategy for mixing in incompressible flows.

physics.flu-dyn↗

arXiv2018

Controlled Layer-by-Layer Oxidation of MoTe2 via O3 Exposure

Growing uniform oxides with various thickness on TMDs is one of the biggest challenges to integrate TMDs into complementary metal oxide semiconductor (CMOS) logic circuits. Here, we report a layer-by-layer oxidation of atomically thin MoTe2 flakes via ozone (O3) exposure. The thickness of MoOx oxide film could be tuning with atomic-level accuracy simply by varying O3 exposure time. Additionally, MoOx-covered MoTe2 shows a hole-dominated transport behavior. Our findings point to a simple and effective strategy for growing homogenous surface oxide film on MoTe2, which is promising for several purposes in metal-oxide-semiconductor transistor, ranging from surface passivation to dielectric layers.

cond-mat.mtrl-sci↗

arXiv2015

The Representation of Line Dirac Delta Function Along a Space Curve

In this paper, we describe the line Dirac delta function of a curve in three-dimensional space in terms of the distance function to the curve. Its extension to level set formulation and plane curves are also developed. The main ideas can be applied for general dimension and codimension.

math.MG↗

arXiv2010

Analysis and simulations of a Viscoelastic Model of Angiogenesis

The work analyzes a one-dimensional viscoelastic model of blood vessel growth under nonlinear friction with surroundings, and provides numerical simulations for various growing cases. For the nonlinear differential equations, two sufficient conditions are proven to guarantee the global existence of biologically meaningful solutions. Examples with breakdown solutions are captured by numerical approximations. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under various biological conditions including blood vessel extension without proliferation and blood vessel regression.

math.AP↗