arXiv subjects

Dongling Wang

Publications and source records attributed to Dongling Wang.

At least 19 recordsLinked to original sources

arXiv2026

A Weighted Integral-Regularized Finite Difference Scheme for the Tempered Fractional Laplacian

The intrinsic singularity of the tempered fractional Laplacian (TFL) remains a major challenge in developing numerical methods that are simultaneously accurate, efficient, and easy to implement. We develop a weighted integral-regularized finite difference (WIRFD) method that regularizes the singular integrand via a multidimensional Taylor expansion incorporating a smooth window function. The resulting integral is decomposed into a regularized term, which is discretized by a punctured trapezoidal rule, and a directly evaluated correction term. For the multidimensional TFL operator, we derive an $O(h^{4-\alpha})$ truncation error bound in the $l^{\infty}$-norm for $\alpha\in(0,2)$ and $u\in C^s(\mathbb{R}^d)$ with $s\geq 8$ by introducing a smooth auxiliary function together with the aliasing formula. For the one-dimensional TFL equation, we establish stability in both the $l^2$- and $l^{\infty}$-norms and optimal $O(h^{4-\alpha})$ convergence for $\alpha\in[1,2)$ based on the strict diagonal dominance of the discrete matrix and a lower bound for its minimum eigenvalue. The Toeplitz structure of the discrete matrix enables FFT-based matrix-vector multiplication, and the resulting linear systems are solved efficiently by a preconditioned conjugate gradient (PCG) method. Numerical experiments corroborate the theoretical results, demonstrating the accuracy, efficiency, and robustness of the proposed method.

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arXiv2026

Regularity and high-order time stepping for semilinear subdiffusion equations with singular initial data beyond the $L^\infty$ framework

This paper aims to analyze a numerical scheme for semilinear subdiffusion problems with singular initial data beyond the $L^\infty$ framework. The main difficulty lies in the stronger singular behavior of the nonlinear term compared with previous analyses. Since the singular initial datum is too rough to guarantee a uniform $L^\infty$ bound for the solution, the usual Lipschitz framework in the base space is no longer sufficient. The analysis must instead be carried out in weaker fractional Sobolev-type spaces, where nonlinear composition is more delicate and the term $f(u(t))$ may exhibit an amplified singularity relative to that of $u(t)$. To overcome this difficulty, we exploit the smoothing properties of the subdiffusion solution operators and formulate suitable nonlinear assumptions in fractional operator spaces. These smoothing estimates allow part of the singularity to be transferred from the nonlinear term to the solution operators, where it can be controlled. Under these assumptions, we establish well-posedness and regularity results for the mild solution and derive a pointwise-in-time error estimate for the exponential convolution quadrature method. Numerical experiments confirm the predicted convergence rates.

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arXiv2026

Average block nonlinear Kaczmarz methods with adaptive momentum for nonlinear systems of equations

The Kaczmarz method is widely recognized as an efficient iterative algorithm for solving large-scale linear systems, owing to its simplicity and low memory requirements. However, the development of its nonlinear extensions for solving large-scale nonlinear systems has seen limited progress. In this work, we introduce a new family of momentum-accelerated averaging block nonlinear Kaczmarz methods tailored for large-scale nonlinear systems and ill-posed problems. Our contributions are twofold: (1) We develop an adaptive strategy for selecting step sizes and momentum coefficients, leading to a new average block nonlinear Kaczmarz method with adaptive momentum (ABNKAm). This algorithm achieves high computational efficiency by requiring only minimal inner-product computations per iteration, which significantly reduces both arithmetic complexity and memory usage. (2) We establish rigorous convergence of the ABNKAm under mild assumptions, proving that the method converges exponentially to the unique solution nearest to the initial point. Moreover, under suitable conditions, we provide a theoretical justification of acceleration of the proposed ABNKAm with momentum. Extensive numerical experiments demonstrate that ABNKAm outperforms existing nonlinear Kaczmarz variants in terms of both iteration count and computational time, with particularly notable gains in large-scale problems.

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arXiv2026

Preserving Conservation Laws in the Time-Evolving Natural Gradient Method via Relaxation and Projection Techniques

Neural networks have demonstrated significant potential in solving partial differential equations (PDEs). While global approaches such as Physics-Informed Neural Networks (PINNs) offer promising capabilities, they often lack inherent temporal causality, which can limit their accuracy and stability for time-dependent problems. In contrast, local training frameworks that progressively update network parameters over time are naturally suited for evolving PDEs. However, a critical challenge remains: many physical systems possess intrinsic invariants -- such as energy or mass -- that must be preserved to ensure physically meaningful solutions. This paper addresses this challenge by enhancing the Time-Evolving Natural Gradient (TENG) method, a recently proposed local training framework. We introduce two complementary techniques: (i) a relaxation algorithm that ensures the target solution $u_{\text{target}}$ preserves both quadratic and general nonlinear invariants of the original system, providing a structure-preserving learning target; and (ii) a projection technique that maps the updated network parameters $θ(t)$ back onto the invariant manifold, ensuring the final neural network solution strictly adheres to the conservation laws. Numerical experiments on the inviscid Burgers equation, Korteweg-de Vries equation, and acoustic wave equation demonstrate that our proposed approach significantly improves conservation properties while maintaining high accuracy.

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arXiv2026

Higher-Order Finite Difference Methods for the Tempered Fractional Laplacian

This paper presents a general framework of high-order finite difference (HFD) schemes for the tempered fractional Laplacian (TFL) based on new generating functions obtained from the discrete symbols. Specifically, for sufficiently smooth functions, the resulting discretizations achieve high-order convergence with orders $p=4, 6, 8$. The discrete operators lead to Toeplitz stiffness matrices, allowing efficient matrix-vector multiplications via fast algorithms. Building on these approximations, HFD methods are formulated for solving TFL equations, and their stability and convergence are rigorously analyzed. Numerical simulations confirm the effectiveness of the proposed methods, showing excellent agreement with the theoretical predictions.

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arXiv2026

Temporal Two-Grid Compact Difference Scheme for Benjamin-Bona-Mahony-Burgers Equation

This paper proposes a temporal two-grid compact difference (TTCD) scheme for solving the Benjamin-Bona-Mahony-Burgers (BBMB) equation with initial and periodic boundary conditions. The method consists of three main steps: first, solving a nonlinear system on a coarse time grid of size $τ_c$; then obtaining a coarse approximation on the fine time grid of size $τ_f$ via linear Lagrange interpolation; and finally solving a linearized scheme on the fine grid to obtain the corrected solution. The TTCD scheme reduces computational cost without sacrificing accuracy. Moreover, using the energy method, we rigorously prove the conservation property, unique solvability, convergence, and stability of the proposed scheme. It is shown that the method achieves convergence of order $\mathcal{O}(τ_c^2 + τ_f^2 + h^4)$ in the maximum norm, where $h$ is space step size. Finally, some numerical experiments are provided to demonstrate the effectiveness and feasibility of the proposed strategy.

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arXiv2025

A note on spectral Monte-Carlo method for fractional Poisson equation on high-dimensional ball

Recently, a class of efficient spectral Monte-Carlo methods was developed in \cite{Feng2025ExponentiallyAS} for solving fractional Poisson equations. These methods fully consider the low regularity of the solution near boundaries and leverage the efficiency of walk-on-spheres algorithms, achieving spectral accuracy. However, the underlying formulation is essentially one-dimensional. In this work, we extend this approach to radial solutions in general high-dimensional balls. This is accomplished by employing a different set of eigenfunctions for the fractional Laplacian and deriving new interpolation formulas. We provide a comprehensive description of our methodology and a detailed comparison with existing techniques. Numerical experiments confirm the efficacy of the proposed extension.

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arXiv2025

Convergence to equilibrium for fully discretizations of nonlocal Cahn-Hilliard equation

The study of long-term dynamics for numerical solutions of nonlinear evolution equations, particularly phase field models, has consistently garnered considerable attention. The Cahn-Hilliard (CH) equation is one of the most important phase field models and is widely applied in materials science. In order to more accurately describe the practical phenomena in material microstructural phase transitions, the Nonlocal Cahn-Hilliard (N-CH) equation incorporates a finite range of spatial nonlocal interactions is introduced, which is a generalization of the classic CH equation. However, compared to its classic counterpart, it is very challenging to investigate the long-term asymptotic behavior of solution to the N-CH equation due to the complexity of the nonlocal integral term and the lack of high-order diffusion term. In this paper, we consider first-order and second-order temporal discretization methods for the N-CH equation, respectively, while utilizing a second-order finite difference method for spatial approximation to construct the energy stable fully discrete numerical schemes. Based on energy stability and the Łojasiewicz inequality, we rigorously prove that the numerical solutions of these fully discrete numerical schemes converge to equilibrium as time goes to infinity.

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arXiv2025

Random Greedy Fast Block Kaczmarz Method for Solving Large-Scale Nonlinear Systems

To efficiently solve large scale nonlinear systems, we propose a novel Random Greedy Fast Block Kaczmarz method. This approach integrates the strengths of random and greedy strategies while avoiding the computationally expensive pseudoinversion of Jacobian submatrices, thus enabling efficient solutions for large scale problems. Our theoretical analysis establishes that the proposed method achieves linear convergence in expectation, with its convergence rates upper bound determined by the stochastic greedy condition number and the relaxation parameter. Numerical experiments confirm that when the Jacobian matrix exhibits a favorable stochastic greedy condition number and an appropriate relaxation parameter is selected, the algorithm convergence is significantly accelerated. As a result, the proposed method outperforms other comparable algorithms in both efficiency and robustness.

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arXiv2025

High-order discretization errors for the Caputo derivative in Hölder spaces

Building upon the recent work of Teso and Plociniczak (2025) regarding L1 discretization errors for the Caputo derivative in Hölder spaces, this study extends the analysis to higher-order discretization errors within the same functional framework. We first investigate truncation errors for the L2 and L1-2 methods, which approximate the Caputo derivative via piecewise quadratic interpolation. Then we generalize the results to arbitrary high-order discretization. Theoretical analyses reveal a unified error structure across all schemes: the convergence order equals the difference between the smoothness degree of the function space and the fractional derivative order, i.e., order of error = degree of smoothness - order of the derivative. Numerical experiments validate these theoretical findings.

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arXiv2025

Fractional coupled Halanay inequality and its applications

This paper introduces a generalized fractional Halanay-type coupled inequality, which serves as a robust tool for characterizing the asymptotic stability of diverse time fractional functional differential equations, particularly those exhibiting Mittag-Leffler type stability. Our main tool is a sub-additive property of Mittag-Leffler function and its optimal asymptotic decay rate estimation. Our results further optimize and improve some existing results in the literature. We illustrate two significant applications of this fractional Halanay-type inequality. Firstly, by combining our results in this work with the positive representation method positive representation of delay differential systems, we establish an asymptotic stability criterion for a category of linear fractional coupled systems with bounded delays. This criterion extends beyond the traditional boundaries of positive system theory, offering a new perspective on stability analysis in this domain. Secondly, through energy estimation, we establish the contractility and dissipativity of a class of time fractional neutral functional differential equations. Our analysis reveals the typical long-term polynomial decay behavior inherent in time fractional evolutionary equations, thereby providing a solid theoretical foundation for subsequent numerical investigations.

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arXiv2025

Adaptive Residual-Driven Newton Solver for Nonlinear Systems of Equations

Newton-type solvers have been extensively employed for solving a variety of nonlinear system of algebraic equations. However, for some complex nonlinear system of algebraic equations, efficiently solving these systems remains a challenging task. The primary reason for this challenge arises from the unbalanced nonlinearities within the nonlinear system. Therefore, accurately identifying and balancing the unbalanced nonlinearities in the system is essential. In this work, we propose a residual-driven adaptive strategy to identify and balance the nonlinearities in the system. The fundamental idea behind this strategy is to assign an adaptive weight multiplier to each component of the nonlinear system, with these weight multipliers increasing according to a specific update rule as the residual components increase, thereby enabling the Newton-type solver to select a more appropriate step length, ensuring that each component in the nonlinear system experiences sufficient reduction rather than competing against each other. More importantly, our strategy yields negligible additional computational overhead and can be seamlessly integrated with other Newton-type solvers, contributing to the improvement of their efficiency and robustness. We test our algorithm on a variety of benchmark problems, including a chemical equilibrium system, a convective diffusion problem, and a series of challenging nonlinear systems. The experimental results demonstrate that our algorithm not only outperforms existing Newton-type solvers in terms of computational efficiency but also exhibits superior robustness, particularly in handling systems with highly imbalanced nonlinearities.

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arXiv2024

Asymptotic stability of many numerical schemes for phase-field modeling

In the recent breakthrough work \cite{xu2023lack}, a rigorous numerical analysis was conducted on the numerical solution of a scalar ODE containing a cubic polynomial derived from the Allen-Cahn equation. It was found that only the implicit Euler method converge to the correct steady state for any given initial value $u_0$ under the unique solvability and energy stability. But all the other commonly used second-order numerical schemes exhibit sensitivity to initial conditions and may converge to an incorrect equilibrium state as $t_n\to\infty$. This indicates that energy stability may not be decisive for the long-term qualitative correctness of numerical solutions. We found that using another fundamental property of the solution, namely monotonicity instead of energy stability, is sufficient to ensure that many common numerical schemes converge to the correct equilibrium state. This leads us to introduce the critical step size constant $h^*=h^*(u_0,ε)$ that ensures the monotonicity and unique solvability of the numerical solutions, where the scaling parameter $ε\in(0,1)$. We prove that the implicit Euler scheme $h^*=h^*(ε)$, which is independent of $u_0$ and only depends on $ε$. Hence regardless of the initial value taken, the simulation can be guaranteed to be correct when $h<h^*$. But for various other numerical methods, no mater how small the step size $h$ is in advance, there will always be initial values that cause simulation errors. In fact, for these numerical methods, we prove that $\inf_{u_0\in \mathbb{R}}h^*(u_0,ε)=0$. Various numerical experiments are used to confirm the theoretical analysis.

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arXiv2024

Mittag-Leffler stability of complete monotonicity-preserving schemes for time-dependent coefficients sub-diffusion equations

A key characteristic of the anomalous sub-solution equation is that the solution exhibits algebraic decay rate over long time intervals, which is often refered to the Mittag-Leffler type stability. For a class of power nonlinear sub-diffusion models with variable coefficients, we prove that their solutions have Mittag-Leffler stability when the source functions satisfy natural decay assumptions. That is the solutions have the decay rate $\|u(t)\|_{L^{s}(Ω)}=O\left( t^{-(α+β)/γ} \right)$ as $t\rightarrow\infty$, where $α$, $γ$ are positive constants, $β\in(-α,\infty)$ and $s\in (1,\infty)$. Then we develop the structure preserving algorithm for this type of model. For the complete monotonicity-preserving ($\mathcal{CM}$-preserving) schemes developed by Li and Wang (Commun. Math. Sci., 19(5):1301-1336, 2021), we prove that they satisfy the discrete comparison principle for time fractional differential equations with variable coefficients. Then, by carefully constructing the fine the discrete supsolution and subsolution, we obtain the long time optimal decay rate of the numerical solution $\|u_{n}\|_{L^{s}(Ω)}=O\left( t_n^{-(α+β)/γ} \right)$ as $t_{n}\rightarrow\infty$, which is fully agree with the theoretical solution. Finally, we validated the analysis results through numerical experiments.

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arXiv2024

Convergence analysis of exponential time differencing scheme for the nonlocal Cahn-Hilliard equation

In this paper, we present a rigorous proof of the convergence of first order and second order exponential time differencing (ETD) schemes for solving the nonlocal Cahn-Hilliard (NCH) equation. The spatial discretization employs the Fourier spectral collocation method, while the time discretization is implemented using ETD-based multistep schemes. The absence of a higher-order diffusion term in the NCH equation poses a significant challenge to its convergence analysis. To tackle this, we introduce new error decomposition formulas and employ the higher-order consistency analysis. These techniques enable us to establish the $\ell^\infty$ bound of numerical solutions under some natural constraints. By treating the numerical solution as a perturbation of the exact solution, we derive optimal convergence rates in $\ell^\infty(0,T;H_h^{-1})\cap \ell^2(0,T; \ell^2)$. We conduct several numerical experiments to validate the accuracy and efficiency of the proposed schemes, including convergence tests and the observation of long-term coarsening dynamics.

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arXiv2022

Optimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations

The solution of the nonlinear initial-value problem $\mathcal{D}_{t}^αy(t)=-λy(t)^γ$ for $t>0$ with $y(0)>0$, where $\mathcal{D}_{t}^α$ is a Caputo derivative of order $α\in (0,1)$ and $λ, γ$ are positive parameters, is known to exhibit $O(t^{α/γ})$ decay as $t\to\infty$. No corresponding result for any discretisation of this problem has previously been proved. In the present paper it is shown that for the class of complete monotonicity-preserving ($\mathcal{CM}$-preserving) schemes (which includes the L1 and Grünwald-Letnikov schemes) on uniform meshes $\{t_n:=nh\}_{n=0}^\infty$, the discrete solution also has $O(t_{n}^{-α/γ})$ decay as $t_{n}\to\infty$. This result is then extended to $\mathcal{CM}$-preserving discretisations of certain time-fractional nonlinear subdiffusion problems such as the time-fractional porous media and $p$-Laplace equations. For the L1 scheme, the $O(t_{n}^{-α/γ})$ decay result is shown to remain valid on a very general class of nonuniform meshes. Our analysis uses a discrete comparison principle with discrete subsolutions and supersolutions that are carefully constructed to give tight bounds on the discrete solution. Numerical experiments are provided to confirm our theoretical analysis.

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arXiv2022

Mittag--Leffler stability of numerical solutions to time fractional ODEs

The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3 (1990), 216-240), we are able to prove that both $\mathcal{L}$1 scheme and strong $A$-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong $A$-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on $α$-difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation and the decay rate of the continuous fractional resolvent operator. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations, fractional linear system and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.

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arXiv2021

Numerical stability of Grünwald-Letnikov method for time fractional delay differential equations

This paper is concerned with the numerical stability of time fractional delay differential equations (F-DDEs) based on Grünwald-Letnikov (GL) approximation (also called fraction backward Euler scheme) for the Caputo fractional derivative, in particular, the numerical stability region and the Mittag-Leffler stability. Using the boundary locus technique, we first derive the exact expression of the numerically stability region in the parameter plane, and show that the fractional backward Euler scheme based on GL scheme is not $τ(0)$-stable, which is different from the backward Euler scheme for integer DDE models. Secondly, we also prove the numerical Mittag-Leffler stability for the numerical solutions provided that the parameters fall into the numerical stability region, by employing the singularity analysis of generating function. Our results show that the numerical solutions of F-DDEs are completely different from the classical integer order DDEs, both in terms of $τ(0)$-stabililty and the long-time decay rate.

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