arXiv subjects

Hengfei Ding

Publications and source records attributed to Hengfei Ding.

9 recordsLinked to original sources

arXiv2025

A Relaxed Step-Ratio Constraint for Time-Fractional Cahn--Hilliard Equations: Analysis and Computation

Numerical solutions of time-fractional differential equations encounter significant challenges arising from solution singularities at the initial time. To address this issue, the construction of nonuniform temporal meshes satisfying $τ_k/τ_{k-1} \geq 1$ has emerged as an effective strategy, where $τ_k$ represents the $k$-th time-step size. For the time-fractional Cahn-Hilliard equation, Liao et al.~[\textit{IMA J. Numer. Anal.}, \textbf{45} (2025), 1425--1454] developed an analytical framework using a variable-step L2 formula with the constraint $0.3960 \leq τ_k/τ_{k-1} \leq r^*(α)$, where $r^*(α) \geq 4.660$ for $α\in (0,1)$. The present work makes substantial theoretical progress by introducing innovative splitting techniques that relax the step-size ratio restriction to $τ_k/τ_{k-1} \leq ρ^*(α)$, with $ρ^*(α) > \overlineρ \approx 4.7476114$. This advancement provides significantly greater flexibility in time-step selection. Building on this theoretical foundation, we propose a refined L2-type temporal approximation coupled with a fourth-order compact difference spatial discretization, yielding an efficient numerical scheme for the time-fractional Cahn-Hilliard problem. Our rigorous analysis establishes the scheme's fundamental properties, including unique solvability, exact discrete volume conservation, proper energy dissipation laws, and optimal convergence rates. For practical implementation, we construct a specialized nonuniform mesh that automatically satisfies the relaxed constraint $ρ^*(α) > 4.7476114$.

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arXiv2025

Implicit integration factor method coupled with Padé approximation strategy for nonlocal Allen-Cahn equation

The space nonlocal Allen-Cahn equation is a famous example of fractional reaction-diffusion equations. It is also an extension of the classical Allen-Cahn equation, which is widely used in physics to describe the phenomenon of two-phase fluid flows.Due to the nonlocality of the nonlocal operator, numerical solutions to these equations face considerable challenges.It is worth noting that whether we use low-order or high-order numerical differential formulas to approximate the operator, the corresponding matrix is always dense, which implies that the storage space and computational cost required for the former and the latter are the same. However, the higher-order formula can significantly improve the accuracy of the numerical scheme.Therefore, the primary goal of this paper is to construct a high-order numerical formula that approximates the nonlocal operator.To reduce the time step limitation in existing numerical algorithms, we employed a technique combining the compact integration factor method with the Padé approximation strategy to discretize the time derivative.A novel high-order numerical scheme, which satisfies both the maximum principle and energy stability for the space nonlocal Allen-Cahn equation, is proposed.Furthermore, we provide a detailed error analysis of the differential scheme, which shows that its convergence order is $\mathcal{O}\left(τ^2+h^6\right)$.Especially, it is worth mentioning that the fully implicit scheme with sixth-order accuracy in spatial has never been proven to maintain the maximum principle and energy stability before.Finally, some numerical experiments are carried out to demonstrate the efficiency of the proposed method.

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arXiv2025

Mathematical analysis and numerical simulation of coupled nonlinear space-fractional Ginzburg-Landau equations

The coupled nonlinear space fractional Ginzburg-Landau (CNLSFGL) equations with the fractional Laplacian have been widely used to model the dynamical processes in a fractal media with fractional dispersion. Due to the existence of fractional power derivatives and strong nonlinearity, it is extremely difficult to mathematically analyze the CNLSFGL equations and construct efficient numerical algorithms. For this reason, this paper aims to investigate the theoretical results about the considered system and construct a novel high-order numerical scheme for this coupled system. We prove rigorously an a priori estimate of the solution to the coupled system and the well-posedness of its weak solution. Then, to develop the efficient numerical algorithm, we construct a fourth-order numerical differential formula to approximate the fractional Laplacian. Based on this formula, we construct a high-order implicit difference scheme for the coupled system. Furthermore, the unique solvability and convergence of the established algorithm are proved in detail. To implement the implicit algorithm efficiently, an iterative algorithm is designed in the numerical simulation. Extensive numerical examples are reported to further demonstrate the correctness of the theoretical analysis and the efficiency of the proposed numerical algorithm.

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arXiv2016

Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations

It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the nonlocal properties of fractional operators. Therefore, developing some high-order numerical approximation formulas for fractional derivatives play a more important role in numerically solving fractional differential equations. This paper focuses on constructing (generalized) high-order fractional-compact numerical approximation formulas for Riesz derivatives. Then we apply the developed formulas to the one- and two-dimension Riesz spatial fractional reaction-dispersion equations. The stability and convergence of the derived numerical algorithms are strictly studied by using the energy analysis method. Finally, numerical simulations are given to demonstrate the efficiency and convergence orders of the presented numerical algorithms.

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arXiv2016

A new second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and its application

Compared to the the classical first-order Gr\"unwald-Letnikov formula at time $t_{k+1} (\textmd{or}\, t_{k})$, we firstly propose a second-order numerical approximate scheme for discretizing the Riemann-Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank-Niclson technique applied to the time-fractional differential equations. The established formula has the following form $$ \begin{array}{lll} \displaystyle \,_{\mathrm{RL}}{{{\mathrm{D}}}}_{0,t}^{\alpha}u\left(t\right)\left|\right._{t=t_{k+\frac{1}{2}}}= \tau^{-\alpha}\sum\limits_{\ell=0}^{k} \varpi_{\ell}^{(\alpha)}u\left(t_k-\ell\tau\right) +\mathcal{O}(\tau^2),\,\,k=0,1,\ldots, \alpha\in(0,1), \end{array} $$ where the coefficients $\varpi_{\ell}^{(\alpha)}$ $(\ell=0,1,\ldots,k)$ can be determined via the following generating function $$ \begin{array}{lll} \displaystyle G(z)=\left(\frac{3\alpha+1}{2\alpha}-\frac{2\alpha+1}{\alpha}z+\frac{\alpha+1}{2\alpha}z^2\right)^{\alpha},\;|z|<1. \end{array} $$ Applying this formula to the time fractional Cable equations with Riemann-liouville derivative in one or two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.

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arXiv2015

High-order numerical algorithms for Riesz derivatives via constructing new generating functions

A class of high-order numerical algorithms for Riesz derivatives are established through constructing new generating functions. Such new high-order formulas can be regarded as the modification of the classical (or shifted) Lubich's difference ones, which greatly improve the convergence orders and stability for time-dependent problems with Riesz derivatives. In rapid sequence, we apply the 2nd-order formula to one-dimension Riesz spatial fractional partial differential equations to establish an unconditionally stable finite difference scheme with convergent order $O(\tau^2+h^2)$, where $\tau$ and $h$ are the temporal and spatial stepsizes, respectively. Finally, some numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the derived numerical algorithms.

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arXiv2015

High-order fractional-compact finite difference method for Riesz spatial telegraph equation

In this paper, we establish even order compact numerical schemes (4th-order, 6th-order, 8th-order, 10th-order) for Riesz derivatives by using the symmetrical fractional centred difference operator. Then we apply the derived 4th-order algorithm to the Riesz spatial telegraph equation. We carefully study the stability and convergence by matrix method, and show that convergence orders in temporal and spatial directions are both 4th order. Numerical experiments are displayed which support the compact difference schemes for Riesz derivatives and the Riesz spatial telegraph equation.

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arXiv2014

High-order Numerical Methods for Riesz Space Fractional Turbulent Diffusion Equation

Numerical methods for fractional calculus attract increasing interests due to its wide applications in various fields such as physics, mechanics, etc. In this paper, we focus on constructing high-order algorithms for Riesz derivatives, where the convergence orders cover from the second order to the sixth order. Then we apply the established schemes to the Riesz space fractional turbulent diffusion equation. Numerical experiments are displayed which support the theoretical analysis.

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arXiv2014

High-order Compact Difference Schemes for the Modified Anomalous Subdiffusion Equation

In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for Riemann-Liouvile derivative is established based on fractional center difference operator. We apply these methods to fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability and convergence analysis of these difference schemes are studied by Fourier method in details. The convergence orders of these numerical schemes are $\mathcal {O}(\tau^2+h^6)$ and $\mathcal {O}(\tau^2+h^8)$, respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis.

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