arXiv subjects

Zhengru Zhang

Publications and source records attributed to Zhengru Zhang.

18 recordsLinked to original sources

arXiv2026

A Second Order Positivity-Preserving Scheme for the Nonlocal Cahn-Hilliard System with Variable Mobility and Logarithmic Flory-Huggins Potential

A second order finite difference scheme is analyzed for the nonlocal Cahn-Hilliard system with variable mobility and the singular Flory-Huggins logarithmic potential. The temporal discretization employs a modified Crank-Nicolson formulation, while the mobility is treated explicitly to guarantee ellipticity and reduce computational cost. To strictly enforce the positivity-preserving property at the discrete level, a nonlinear regularization term is added into the numerical scheme. Rigorous analysis shows that the scheme is uniquely solvable and stable under a modified numerical energy. The well-known analytical challenge associated with the variable mobility has to be handled carefully. To overcome this, a convergence framework is constructed based on two non-standard techniques: (1) a higher-order asymptotic expansion (extending up to the third order in time and fourth order in space) to retain a sufficiently order of accuracy; (2) a two-stage error strategy, wherein a rough error estimate first guarantees the discrete boundedness of the mobility and nonlinear terms, followed by a refined error analysis that yields the optimal convergence rate. Numerical experiments validate the theoretical results and demonstrate the robustness of the proposed scheme.

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arXiv2026

Nonlocal modeling of opinion alignment and environmental feedback: Spatial aggregation and non-consensus patterns

The formation of public opinion in modern information environments is shaped by the interplay between social conformity and information exposure. While social interactions promote opinion alignment, heterogeneous visibility and selective exposure may reinforce local agreement, a mechanism commonly associated with the echo chamber effects. To describe how such reinforcement influences spatially heterogeneous opinion activity and non-consensus patterns, we propose a spatial opinion dynamics model with attention-mediated feedback. The model couples nonlocal alignment with an evolving attention field and captures a self-reinforcing mechanism in which regions of high opinion activity attract greater visibility. Starting from agent-based jump mechanism inspired by bounded confidence interactions and biased random walks induced by environments, we formally derive a nonlocal advection-cross-diffusion system, where opinion transport is driven by nonlocal conformity and modulated by attention-dependent redistribution. We characterize the transition from spatially homogeneous consensus states to non-consensus clustered regimes through linear stability analysis of the homogeneous equilibrium. The results show that attention-mediated feedback has an explicit correction on the instability threshold and enlarges the parameter regime in which clustering occurs, thereby promoting persistent spatial heterogeneity and non-consensus patterns. Numerical simulations based on a structure-preserving IMEX spectral method support the theoretical predictions and quantify the resulting aggregation phenomena. These findings provide a macroscopic description of how nonlocal alignment and environmental feedback jointly shape spatial signatures of non-consensus patterns.

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arXiv2026

Bound preserving and mass conservative methods for the nonlocal Cahn-Hilliard equation with the logarithmic Flory-Huggins potential

It is well known that the exponential time differencing (ETD) method has been successfully applied to the classic Cahn-Hilliard equation with double well potential. However, this numerical method can not be extended to the Cahn-Hilliard equation with Flory-Huggins potential directly due to the fact that the the numerical solution may go beyond the physical interval which leads the non-physical solution. In this paper, we develop and analyze first- and second-order numerical schemes for the nonlocal Cahn-Hilliard equation with the classic Flory-Huggins energy potential. In more detail, the ETD method is firstly used to obtain the prediction solution, and then this prediction solution is corrected by the projection method to avoid non-physical solution. The proposed method is shown to preserve bound and mass conservation in discrete settings. In addition, error estimates for the numerical solution are rigorously obtained for both schemes. Extensive numerical tests and comparisons are conducted to demonstrate the performance of the proposed schemes.

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arXiv2026

Efficient structure-preserving scheme for chemotaxis PDE system with singular sensitivity in crime and epidemic modeling

The chemotaxis PDE system with singular sensitivity was originally proposed by Short et al. (Math. Mod. Meth. Appl. Sci., 2008) as the continuum limit of a biased random walk model to account for the formation of crime hotspots and environmental feedback successfully. Recently, this idea has also been applied to epidemiology to model the impact of human social behaviors on disease transmission. In order to characterize the phase transition, pattern formation and statistical properties in the long-term dynamics, a stable and accurate numerical scheme is urgently demanded, which still remains challenging due to the positivity constraint on the singular sensitivity and the absence of an energy functional. In particular, the loss of positivity may produce nonphysical states and even cause spurious blow-up. To address these numerical challenges, this paper constructs an efficient positivity-preserving, implicit-explicit scheme with second-order accuracy. A rigorous error estimation is provided with the Lagrange multiplier correction to deal with the singular sensitivity. The whole framework is extended to a multi-agent epidemic model with degenerate diffusion, in which both positivity and mass conservation are achieved. Numerical experiments are performed to validate the theoretical results and demonstrate the necessity of the correction strategy. Our simulations reveal rich dynamical behaviors, including the phase transition between aggregation-dominated and dissipative regimes, as well as the nucleation, spread, and dissipation of crime hotspots. For the epidemic model, the results further show that spatial clustering of population density may accelerate virus transmission and significantly amplify the infectious wave.

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arXiv2025

Convergence analysis of positivity-preserving finite difference scheme for the Flory-Huggins-Cahn-Hilliard equation with dynamical boundary condition

The Cahn-Hilliard equation has a wide range of applications in many areas of physics and chemistry. To describe the short-range interaction between the solution and the boundary, scientists have constructed dynamical boundary conditions by introducing boundary energy. In this work, the dynamical boundary condition is located on two opposite edges of a square domain and is connected with bulk by a normal derivative. A convex-splitting numerical approach is proposed to enforce the positivity-preservation and energy dissipation, combined with the finite difference spatial approximation. The $\ell^\infty(0,T;H_h^{-1}) \cap \ell^2(0,T;H_h^1)$ convergence analysis and error estimate is theoretically established, with the first order accuracy in time and second order accuracy in space. The bulk and surface discrete mass conservation of the exact solution is required to reach the mean-zero property of the error function, so that the associated discrete $H_h^{-1}$ norm is well-defined. The mass conservation on the physical boundary is maintained by the classic Fourier projection. In terms of the mass conservation in bulk, we introduce a trigonometric auxiliary function based on the truncation error expansion, so that the bulk mass conservation is achieved, and it has no effect on the boundary. The smoothness of trigonometric function makes the Taylor expansion valid and maintains the convergence order of truncation error as well. As a result, the convergence analysis could be derived with a careful nonlinear error estimate.

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arXiv2025

A uniquely solvable and positivity-preserving finite difference scheme for the Flory-Huggins-Cahn-Hilliard equation with dynamical boundary condition

In this paper we propose and analyze a finite difference numerical scheme for the Flory-Huggins-Cahn-Hilliard equation with dynamical boundary condition. The singular logarithmic potential is included in the Flory-Huggins energy expansion. Meanwhile, a dynamical evolution equation for the boundary profile corresponds to a lower-dimensional singular energy potential. In turn, a theoretical analysis for the coupled system becomes very challenging, since it contains nonlinear and singular energy potentials for both the interior region and on the boundary. In the numerical design, a convex splitting approach is applied to the chemical potential associated with the energy both at the interior region and on the boundary: implicit treatments for the singular and logarithmic terms, as well as the surface diffusion terms, combined with an explicit treatment for the concave expansive term. In addition, the discrete boundary condition for the phase variable is coupled with the evolutionary equation of the boundary profile. The resulting numerical system turns out to be highly nonlinear, singular and coupled. A careful finite difference approximation and convexity analysis reveals that such a numerical system could be represented as a minimization of a discrete numerical energy functional, which contains both the interior and boundary integrals. More importantly, all the singular terms correspond to a discrete convex functional. As a result, a unique solvability and positivity-preserving analysis could be theoretically justified, based on the subtle fact that the singular nature of the logarithmic terms around the singular limit values prevent the numerical solutions reaching these values. The total energy stability analysis could be established by a careful estimate over the finite difference inner product. Some numerical results are presented in this article.

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arXiv2025

A positivity-preserving, second-order energy stable and convergent numerical scheme for a ternary system of macromolecular microsphere composite hydrogels

A second order accurate numerical scheme is proposed and analyzed for the periodic three-component Macromolecular Microsphere Composite(MMC) hydrogels system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free energy potential. This numerical scheme with energy stability is based on the Backward Differentiation Formula(BDF) method in time derivation combining with Douglas-Dupont regularization term, combined the finite difference method in space. We provide a theoretical justification of positivity-preserving property for all the singular terms, i.e., not only the two phase variables are always between $0$ and $1$, but also the sum of the two phase variables is between $0$ and $1$, at a point-wise level. In addition, an optimal rate convergence analysis is provided in this paper, in which a higher order asymptotic expansion of the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This paper will be the first to combine the following theoretical properties for a second order accurate numerical scheme for the ternary MMC system: (i) unique solvability and positivity-preserving property; (ii) energy stability; (iii) and optimal rate convergence. A few numerical results are also presented.

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arXiv2024

A structure-preserving implicit exponential time differencing scheme for Maxwell-Amp`ere Nernst-Planck model

The transport of charged particles, which can be described by the Maxwell-Ampere Nernst-Planck (MANP) framework, is essential in various applications including ion channels and semiconductors. We propose a decoupled structure-preserving numerical scheme for the MANP model in this work. The Nernst-Planck equations are treated by the implicit exponential time differencing method associated with the Slotboom transform to preserve the positivity of the concentrations. In order to be effective with the Fast Fourier Transform, additional diffusive terms are introduced into Nernst-Planck equations. Meanwhile, the correction is introduced in the Maxwell-Ampere equation to fulfill Gauss's law. The curl-free condition for electric displacement is realized by a local curl-free relaxation algorithm whose complexity is O(N). We present sufficient restrictions on the time and spatial steps to satisfy the positivity and energy dissipation law at a discrete level. Numerical experiments are conducted to validate the expected numerical accuracy and demonstrate the structure-preserving properties of the proposed method.

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arXiv2023

Convergence analysis of a positivity-preserving numerical scheme for the Cahn-Hilliard-Stokes system with Flory-Huggins energy potential

A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. A convex splitting is applied to the chemical potential, which in turns leads to the implicit treatment for the singular logarithmic terms and the surface diffusion term, and an explicit update for the expansive concave term. The convective term for the phase variable, as well as the coupled term in the Stokes equation, are approximated in a semi-implicit manner. In the spatial discretization, the marker and cell (MAC) difference method is applied, which evaluates the velocity components, the pressure and the phase variable at different cell locations. Such an approach ensures the divergence-free feature of the discrete velocity, and this property plays an important role in the analysis. The positivity-preserving property and the unique solvability of the proposed numerical scheme are theoretically justified, utilizing the singular nature of the logarithmic term as the phase variable approaches the singular limit values. An unconditional energy stability analysis is standard, as an outcome of the convex-concave decomposition technique. A convergence analysis with accompanying error estimate is provided for the proposed numerical scheme. In particular, a higher order consistency analysis, accomplished by supplementary functions, is performed to ensure the separation properties of numerical solution. In turn, using the approach of rough and refined error (RRE) estimates, we are able to derive an optimal rate convergence. To conclude, several numerical experiments are presented to validate the theoretical analysis.

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arXiv2022

Second order stabilized semi-implicit scheme for the Cahn-Hilliard model with dynamic boundary conditions

We study the numerical algorithm and error analysis for the Cahn-Hilliard equation with dynamic boundary conditions. A second-order in time, linear and energy stable scheme is proposed, which is an extension of the first-order stabilized approach. The corresponding energy stability and convergence analysis of the scheme are derived theoretically. Some numerical experiments are performed to verify the effectiveness and accuracy of the second-order numerical scheme, including numerical simulations under various initial conditions and energy potential functions, and comparisons with the literature works.

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arXiv2021

A positivity-preserving, energy stable scheme for a Ternary Cahn-Hilliard system with the singular interfacial parameters

In this paper, we construct and analyze a uniquely solvable, positivity preserving and unconditionally energy stable finite-difference scheme for the periodic three-component Macromolecular Microsphere Composite (MMC) hydrogels system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free energy potential. The proposed scheme is based on a convex-concave decomposition of the given energy functional with two variables, and the centered difference method is adopted in space. We provide a theoretical justification that this numerical scheme has a pair of unique solutions, such that the positivity is always preserved for all the singular terms, i.e., not only two phase variables are always between $0$ and $1$, but also the sum of two phase variables is between $0$ and $1$, at a point-wise level. In addition, we use the local Newton approximation and multigrid method to solve this nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, energy dissipation and mass conservation properties.

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arXiv2021

An energy stable finite element scheme for the three-component Cahn-Hilliard-type model for macromolecular microsphere composite hydrogels

In this article, we present and analyze a finite element numerical scheme for a three-component macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn-Hilliard-type equation with Flory-Huggins-deGennes energy potential. The numerical approach is based on a convex-concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme.

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arXiv2021

A positivity-preserving and convergent numerical scheme for the binary fluid-surfactant system

In this paper, we develop a first order (in time) numerical scheme for the binary fluid surfactant phase field model. The free energy contains a double-well potential, a nonlinear coupling entropy and a Flory-Huggins potential. The resulting coupled system consists of two Cahn-Hilliard type equations. This system is solved numerically by finite difference spatial approximation, in combination with convex splitting temporal discretization. We prove the proposed scheme is unique solvable, positivity-preserving and unconditionally energy stable. In addition, an optimal rate convergence analysis is provided for the proposed numerical scheme, which will be the first such result for the binary fluid-surfactant system. Newton iteration is used to solve the discrete system. Some numerical experiments are performed to validate the accuracy and energy stability of the proposed scheme.

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arXiv2020

Structure-preserving, energy stable numerical schemes for a liquid thin film coarsening model

In this paper, two finite difference numerical schemes are proposed and analyzed for the droplet liquid film model, with a singular Leonard-Jones energy potential involved. Both first and second order accurate temporal algorithms are considered. In the first order scheme, the convex potential and the surface diffusion terms are implicitly, while the concave potential term is updated explicitly. Furthermore, we provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity is always preserved for the phase variable at a point-wise level, so that a singularity is avoided in the scheme. In fact, the singular nature of the Leonard-Jones potential term around the value of 0 prevents the numerical solution reaching such singular value, so that the positivity structure is always preserved. Moreover, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. In the second order numerical scheme, the BDF temporal stencil is applied, and an alternate convex-concave decomposition is derived, so that the concave part corresponds to a quadratic energy. In turn, the combined Leonard-Jones potential term is treated implicitly, and the concave part the is approximated by a second order Adams-Bashforth explicit extrapolation, and an artificial Douglas-Dupont regularization term is added to ensure the energy stability. The unique solvability and the positivity-preserving property for the second order scheme could be similarly established. In addition, optimal rate convergence analysis is provided for both the first and second order accurate schemes. A few numerical simulation results are also presented, which demonstrate the robustness of the numerical schemes.

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arXiv2020

A positivity-preserving second-order BDF scheme for the Cahn-Hilliard equation with variable interfacial parameters

We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method time derivation combining with Douglas-Dupont regularization term. In addition, we present a point-wise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$ norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.

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arXiv2017

High efficiently numerical simulation of the TDGL equation with reticular free energy in hydrogel

In this paper, we focus on the numerical simulation of phase separation about macromolecule microsphere composite (MMC) hydrogel. The model equation is based on Time-Dependent Ginzburg-Landau (TDGL) equation with reticular free energy. We have put forward two $L^2$ stable schemes to simulate simplified TDGL equation. In numerical experiments, we observe that simulating the whole process of phase separation requires a considerably long time. We also notice that the total free energy changes significantly in initial time and varies slightly in the following time. Based on these properties, we introduce an adaptive strategy based on one of stable scheme mentioned. It is found that the introduction of the time adaptivity cannot only resolve the dynamical changes of the solution accurately but also can significantly save CPU time for the long time simulation.

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arXiv2017

A Second-Order Energy Stable Backward Differentiation Formula Method for the Epitaxial Thin Film Equation with Slope Selection

In this paper, we study a novel second-order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for the higher oder in time temporal discretization is how to ensure an unconditional energy stability and an efficient numerical implementation. We propose a general framework for designing the higher order in time numerical scheme with unconditional energy stability by using the BDF method with constant coefficient stabilized terms. Based on the unconditional energy stability property, we derive an $L^\infty_h (0,T; H_{h}^2)$ stability for the numerical solution and provide an optimal the convergence analysis. To deal with the 4-Laplacian solver in an $L^{2}$ gradient flow at each time step, we apply an efficient preconditioned steepest descent algorithm and preconditioned nonlinear conjugate gradient algorithm to solve the corresponding nonlinear system. Various numerical simulations are present to demonstrate the stability and efficiency of the proposed schemes and slovers.

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arXiv2016

Convergence Analysis and Numerical Implementation of a Second Order Numerical Scheme for the Three-Dimensional Phase Field Crystal Equation

In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed in [46], with the unique solvability and unconditional energy stability established. However, its convergence analysis remains open. We present a detailed convergence analysis in this article, in which the maximum norm estimate of the numerical solution over grid points plays an essential role. Moreover, we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain, and various three-dimensional numerical results are presented, including the numerical convergence test, complexity test of the multigrid solver and the polycrystal growth simulation.

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