arXiv subjects

Fukeng Huang

Publications and source records attributed to Fukeng Huang.

8 recordsLinked to original sources

arXiv2026

A new class of efficient linear higher-order schemes for the Landau-Lifshitz-Gilbert equation with reduced restriction on the damping parameter

Classical high-order backward differentiation formula (BDF) methods for the Landau-Lifshitz-Gilbert (LLG) equation often suffer from restrictive stability constraints, requiring small time steps and imposing stringent lower bounds on the damping parameter. These limitations become particularly severe for schemes of order higher than three. In this paper, we develop a class of high-order generalized BDF (GBDF) schemes for the LLG equation, including both semi-implicit and fully explicit treatments of the gyromagnetic term. The proposed schemes significantly improve stability properties and substantially relax the damping parameter constraints, but introduce essential difficulty in its analysis compared to the classical BDF schemes. We construct a novel multiplier which enables us to carry out a energy-based error analysis. This approach yields optimal-order error estimates under considerably weaker assumptions on the damping parameter than those required for classical BDF schemes. Numerical experiments are presented to confirm the theoretical results, and demonstrate that the proposed GBDF schemes achieve higher accuracy, enhanced stability, and much wider admissible damping regimes compared to classical high-order BDF methods.

math.NA↗

arXiv2026

High-order, long-time stable and parallel decoupled GBDF$k$ SAV ensemble schemes for the Navier--Stokes--Darcy flow with random hydraulic conductivity tensors

We develop and analyze high-order ensemble schemes for the unsteady Navier--Stokes--Darcy system with uncertain initial conditions, forcing terms, hydraulic conductivity tensors, and Lions-Beavers-Joseph-Saffman interface conditions. The proposed schemes which are called GSAV-GBDF$k$-Ensemble schemes integrate a partitioned decoupling strategy, the generalized scalar auxiliary variable (GSAV) approach, and generalized BDF$k$ discretizations. This framework achieves high-order temporal accuracy and long-time stability, permits explicit treatment of the nonlinear term, and facilitates an efficient ensemble implementation for multiple parameter realizations by sharing a single, unified coefficient matrix at each time step. Moreover, the numerical solutions are shown to satisfy uniform-in-time bounds without time-step restrictions. Owing to the ensemble formulation, the resulting linear systems share common coefficient matrices, which significantly improves computational efficiency. We further establish optimal-order error estimates for the proposed high-order schemes. Numerical results are included to confirm the theoretical analysis and to illustrate the accuracy, stability, and efficiency of the proposed methods.

math.NA↗

arXiv2025

Stability and error analysis of a new class of higher-order consistent splitting schemes for the Navier-Stokes equations

A new class of fully decoupled consistent splitting schemes for the Navier-Stokes equations are constructed and analyzed in this paper. The schemes are based on the Taylor expansion at $t^{n+\beta}$ with $\beta\ge 1$ being a free parameter. It is shown that by choosing {\color{black} $\beta= 3, \,6,\,9$} respectively for the second-, third- and fourth-order schemes, their numerical solutions are uniformed bounded in a strong norm, and admit optimal global-in-time convergence rates in both 2D and 3D. {\color{black}These } results are the first stability and convergence results for any fully decoupled, higher than second-order schemes for the Navier-Stokes equations. Numerical results are provided to show that the third- and fourth-order schemes based on the usual BDF (i.e. $\beta=1$) are not unconditionally stable while the new third- and fourth-order schemes with suitable $\beta$ are unconditionally stable and lead to expected convergence rates.

math.NA↗

arXiv2024

On a new class of BDF and IMEX schemes for parabolic type equations

When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit (IMEX) schemes for parabolic type equations based on the Taylor expansions at time $t^{n+\beta}$ with $\beta > 1$ being a tunable parameter. These new schemes, with a suitable $\beta$, allow larger time steps at higher-order for stiff problems than that is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes, and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.

math.NA↗

arXiv2023

Phase-field simulations for dripping-to-jetting transitions: Effects of low interfacial tension and bulk diffusion

The dripping-to-jetting transitions in coaxial flows have been experimentally well studied for systems of high interfacial tension, where the capillary number of the outer fluid and the Weber number of the inner fluid are in control. Recent experiments have shown that in systems of low interfacial tension, the transitions driven by the inner flow are no longer dominated by the inertial force alone, and the viscous drag force due to the inner flow is also quantitatively important. In the present work, we carry out numerical simulations based on the Cahn-Hilliard-Navier-Stokes model, aiming for a more complete and quantitative study that is needed for understanding the effects of interfacial tension when it becomes sufficiently low. The Cahn-Hilliard-Navier-Stokes model is solved by using an accurate and efficient spectral method in a cylindrical domain with axisymmetry, and numerical results obtained for jet and drop radii demonstrate the accuracy of our computation. Plenty of numerical examples are systematically presented to show the dripping-to-jetting transitions driven by the outer flow and inner flow respectively. In particular, for transitions dominated by inner flow, detailed results reveal how the magnitude of interfacial tension quantitatively determines the relative importance of the inertial and viscous forces due to the inner flow at the transition point. Our numerical results are found to be consistent with the experimental observation. Finally, the degree of bulk diffusion is varied to investigate its quantitative effect on the condition for the occurrence of transition. Such effect is expected for systems of ultralow interfacial tension where interfacial motion is more likely to be driven by bulk diffusion.

physics.flu-dyn↗

arXiv2022

Diffuse-interface approach to competition between viscous flow and diffusion in pinch-off dynamics

The pinch-off dynamics of a liquid thread has been studied through numerical simulations and theoretical analysis. Occurring at small length scales, the pinch-off dynamics admits similarity solutions that can be classified into the Stokes regime and the diffusion-dominated regime, with the latter being recently experimentally observed in aqueous two-phase systems [Phys. Rev. Lett. 123, 134501 (2019)]. Derived by applying Onsager's variational principle, the Cahn-Hilliard-Navier-Stokes model is employed as a minimal model capable of describing the interfacial motion driven by not only advection but also diffusion. By analyzing the free energy dissipation mechanisms in the model, a characteristic length scale is introduced to measure the competition between diffusion and viscous flow in interfacial motion. This length scale is typically of nanometer scale for systems far from the critical point, but can approach micrometer scale for aqueous two-phase systems close to the critical point. The Cahn-Hilliard-Navier-Stokes model is solved by using an accurate and efficient spectral method in a cylindrical domain with axisymmetry. Ample numerical examples are presented to show the pinch-off processes in the Stokes regime and the diffusion-dominated regime respectively. In particular, the crossover between these two regimes is investigated numerically and analytically to reveal how the scaling behaviors of similarity solutions are to be qualitatively changed as the characteristic length scale is inevitably accessed by the pinching neck of the interface. Discussions are also provided for numerical examples that are performed for the breakup of long liquid filaments and show qualitatively different phenomena in different scaling regimes.

physics.flu-dyn↗

arXiv2021

Stability and error analysis of a class of high-order IMEX schemes for Navier-stokes equations with periodic boundary conditions

We construct high-order semi-discrete-in-time and fully discrete (with Fourier-Galerkin in space) schemes for the incompressible Navier-Stokes equations with periodic boundary conditions, and carry out corresponding error analysis. The schemes are of implicit-explicit type based on a scalar auxiliary variable (SAV) approach. It is shown that numerical solutions of these schemes are uniformly bounded without any restriction on time step size. These uniform bounds enable us to carry out a rigorous error analysis for the schemes up to fifth-order in a unified form, and derive global error estimates in $l^\infty(0,T;H^1)\cap l^2(0,T;H^2)$ in the two dimensional case as well as local error estimates in $l^\infty(0,T;H^1)\cap l^2(0,T;H^2)$ in the three dimensional case. We also present numerical results confirming our theoretical convergence rates and demonstrating advantages of higher-order schemes for flows with complex structures in the double shear layer problem.

math.NA↗

arXiv2021

Implicit-explicit BDF $k$ SAV schemes for general dissipative systems and their error analysis

We construct efficient implicit-explicit BDF$k$ scalar auxiliary variable (SAV) schemes for general dissipative systems. We show that these schemes are unconditionally stable, and lead to a uniform bound of the numerical solution in the norm based on the principal linear operator in the energy. Based on this uniform bound, we carry out a rigorous error analysis for the $k$th-order $(k=1,2,3,4,5)$ SAV schemes in a unified form for a class of typical Allen-Cahn type and Cahn-Hilliard type equations. We also present numerical results confirming our theoretical convergence rates.

math.NA↗