arXiv subjects

Zengyan Zhang

Publications and source records attributed to Zengyan Zhang.

7 recordsLinked to original sources

arXiv2026

A Geometric Local Parameterization Method for Generalized Hele-Shaw Free Boundary Problems with Source Terms

We develop a meshfree numerical framework for Hele--Shaw free boundary problems with surface tension and source terms based on geometric local parameterization and boundary integral methods. By decomposing the pressure into a particular solution and a harmonic component, the problem is reformulated into a boundary-only system, avoiding volumetric meshing of the evolving domain. For general source terms, we propose an eigenfunction-based approximation on a fixed domain and establish error estimates for both the truncation and coefficient approximation. Numerical experiments verify the accuracy and convergence of the proposed method, and an application to a tumor growth model demonstrates its effectiveness for coupled moving-boundary problems.

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arXiv2026

Geometric local parameterization for solving Hele-Shaw problems with surface tension

In this work, we introduce a novel computational framework for solving the two-dimensional Hele-Shaw free boundary problem with surface tension. The moving boundary is represented by point clouds, eliminating the need for a global parameterization. Our approach leverages Generalized Moving Least Squares (GMLS) to construct local geometric charts, enabling high-order approximations of geometric quantities such as curvature directly from the point cloud data. This local parameterization is systematically employed to discretize the governing boundary integral equation, including an analytical formula of the singular integrals. We provide a rigorous convergence analysis for the proposed spatial discretization, establishing consistency and stability under certain conditions. The resulting error bound is derived in terms of the size of the uniformly sampled point cloud data on the moving boundary, the smoothness of the boundary, and the order of the numerical quadrature rule. Numerical experiments confirm the theoretical findings, demonstrating high-order spatial convergence and the expected temporal convergence rates. The method's effectiveness is further illustrated through simulations of complex initial shapes, including interfaces driven by anisotropic surface tension, which correctly evolve towards circular equilibrium states under the influence of surface tension, highlighting the versatility of the method for complex geometry-dependent interface dynamics.

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arXiv2023

General Numerical Framework to Derive Structure Preserving Reduced Order Models for Thermodynamically Consistent Reversible-Irreversible PDEs

In this paper, we propose a general numerical framework to derive structure-preserving reduced order models for thermodynamically consistent PDEs. Our numerical framework has two primary features: (a) a systematic way to extract reduced order models for thermodynamically consistent PDE systems while maintaining their inherent thermodynamic principles and (b) a strategic process to devise accurate, efficient, and structure-preserving numerical algorithms to solve the forehead reduced-order models. The platform's generality extends to various PDE systems governed by embedded thermodynamic laws. The proposed numerical platform is unique from several perspectives. First, it utilizes the generalized Onsager principle to transform the thermodynamically consistent PDE system into an equivalent one, where the transformed system's free energy adopts a quadratic form of the state variables. This transformation is named energy quadratization (EQ). Through EQ, we gain a novel perspective on deriving reduced order models. The reduced order models derived through our method continue to uphold the energy dissipation law. Secondly, our proposed numerical approach automatically provides numerical algorithms to discretize the reduced order models. The proposed algorithms are always linear, easy to implement and solve, and uniquely solvable. Furthermore, these algorithms inherently ensure the thermodynamic laws. In essence, our platform offers a distinctive approach to derive structure-preserving reduced-order models for a wide range of PDE systems abiding by thermodynamic principles.

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arXiv2021

A Remark on the Invariant Energy Quadratization (IEQ) Method for Preserving the Original Energy Dissipation Laws

In this letter, we revisit the IEQ method and provide a new perspective on its ability to preserve the original energy dissipation laws. The invariant energy quadratization (IEQ) method has been widely used to design energy stable numerical schemes for phase-field or gradient flow models. Although there are many merits of the IEQ method, one major disadvantage is that the IEQ method usually respects a modified energy law, where the modified energy is expressed in the auxiliary variables. Still, the dissipation laws in terms of the original energy are not guaranteed. Using the widely-used Cahn-Hilliard equation as an example, we demonstrate that the Runge-Kutta IEQ method indeed can preserve the original energy dissipation laws for certain situations up to arbitrary high-order accuracy. Interested readers are highly encouraged to apply our idea to other phase-field equations or gradient flow models.

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arXiv2021

Improving the Accuracy and Consistency of the Scalar Auxiliary Variable (SAV) Method with Relaxation

The scalar auxiliary variable (SAV) method was introduced by Shen et al. and has been broadly used to solve thermodynamically consistent PDE problems. By utilizing scalar auxiliary variables, the original PDE problems are reformulated into equivalent PDE problems. The advantages of the SAV approach, such as linearity, unconditionally energy stability, and easy-to-implement, are prevalent. However, there is still an open issue unresolved, i.e., the numerical schemes resulted from the SAV method preserve a "modified" energy law according to the auxiliary variables instead of the original variables. Truncation errors are introduced during numerical calculations so that the numerical solutions of the auxiliary variables are no longer equivalent to their original continuous definitions. In other words, even though the SAV scheme satisfies a modified energy law, it does not necessarily satisfy the energy law of the original PDE models. This paper presents one essential relaxation technique to overcome this issue, which we named the relaxed-SAV (RSAV) method. Our RSAV method penalizes the numerical errors of the auxiliary variables by a relaxation technique. In general, the RSAV method keeps all the advantages of the baseline SAV method and improves its accuracy and consistency noticeably. Several examples have been presented to demonstrate the effectiveness of the RSAV approach.

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arXiv2016

Qualitative studies of advective competition system with Beddington-DeAngelis functional response

This paper investigates a reaction-advection-diffusion system modeling interspecific competition between two species over bounded domains. The kinetic terms are assumed to satisfy the Beddington-DeAngelis functional responses. We consider the situation that first species disperse by a combination of random walk and directed movement along the population density of the second species which disperse randomly within the habitat. For multi-dimensional bounded domains, we prove the global existence and boundedness of time-dependent solutions. For one-dimensional finite domains, we study the effect of diffusion and advection on the existence and stability of nonconstant positive steady states to the strongly coupled elliptic system. In particular, our stability result of these nontrivial steady states provides a selection mechanism for stable wavemodes of the time-dependent system. In the limit of diffusion rates, we show that the steady states of this full elliptic system can be approximated by nonconstant positive solutions of a shadow system and then we construct boundary spike solutions to this shadow system. For the full elliptic system, we also investigate solutions with a single boundary spike or an inverted boundary spike, i.e., the first species concentrate on the boundary point while the second species dominate the whole habitat except the boundary point. These spatial structures can be used to model the spatial segregation phenomenon through interspecific competitions. Some numerical studies are performed to illustrate and support our theoretical findings.

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arXiv2016

Pattern formation in a Keller-Segel chemotaxis model with logistic growth

In this paper we investigate pattern formation in Keller--Segel chemotaxis models over a multi--dimensional bounded domain subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as chemoattraction rate $χ$ increases. Then using Crandall--Rabinowitz local theory with $χ$ being the bifurcation parameter, we obtain the existence of nonhomogeneous steady states of the system which bifurcate from this homogeneous steady state. Stability of the bifurcating solutions is also established through rigorous and detailed calculations. Our results provide a selection mechanism of stable wavemode which states that the only stable bifurcation branch must have a wavemode number that minimizes the bifurcation value. Finally we perform extensive numerical simulations on the formation of stable steady states with striking structures such as boundary spikes, interior spikes, stripes, etc. These nontrivial patterns can model cellular aggregation that develop through chemotactic movements in biological systems.

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