arXiv subjects

Weiping Bu

Publications and source records attributed to Weiping Bu.

10 recordsLinked to original sources

arXiv2026

Finite difference methods for three kinds of reaction-diffusion equations with free boundaries

This work considers to numerically solve three kinds of reaction-diffusion equations with free boundaries. First, the popular front-fixing method is used to transform the considered free boundary problems into fixed boundary problems. Then, by employing the finite difference method, numerical schemes with $\mathrm{M}$-matrices as their coefficient matrices are developed for these transformed fixed-boundary problems. Next, for the developed numerical schemes, we establish numerical theory involving positivity preservation, monotonicity preservation, and stability. It is noteworthy that, unlike some existing works that impose tight restrictions on the time step-size to discuss the numerical theory, the proposed numerical schemes require only a mild restriction. Finally, numerical examples are provided to test the developed numerical schemes and to confirm the theoretical results.

math.NA↗

arXiv2025

Numerical simulation of the dual-phase-lag heat conduction equation on a one-dimensional unbounded domain using artificial boundary condition

This paper focuses on the numerical solution of a dual-phase-lag heat conduction equation on a space unbounded domain. First, based on the Laplace transform and the Padé approximation, a high-order local artificial boundary condition is constructed for the considered problem, which effectively transforms the original problem into an initial-boundary value problem on a bounded computational domain. Subsequently, for the resulting reduced problem on the bounded domain equipped with high-order local artificial boundary, a stability result based on the $L^2$-norm is derived. Next, we develop finite difference method for the reduced problem by introducing auxiliary variable to reduce the order of time derivative. The numerical analysis demonstrates that the developed numerical scheme is unconditionally stable and possesses a second-order convergence rate in both space and time. Finally, numerical results are presented to validate the effectiveness of the proposed numerical method and the correctness of the theoretical analysis.

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arXiv2025

Finite element method for a constant time delay subdiffusion equation with Riemann-Liouville fractional derivative

This work considers to numerically solve a subdiffusion equation involving constant time delay $τ$ and Riemann-Liouville fractional derivative. First, a fully discrete finite element scheme is developed for the considered problem under the symmetric graded time mesh, where the Caputo fractional derivative is approximated via the L1 formula, while the Riemann-Liouville integral is discretized using the fractional right rectangular rule. Under the assumption that the exact solution has low regularities at $t=0$ and $τ$, the local truncation errors of both the L1 formula and the fractional right rectangular rule are analyzed. It is worth noting that, by setting the mesh parameter $r=1$, the symmetric graded time mesh will degenerate to a uniform mesh. Consequently, we proceed to discuss the stability and convergence of the proposed numerical scheme under two scenarios. For the uniform time mesh, by introducing a discrete sequence $\{P_k\}$, the unconditional stability and local time error estimate for the developed scheme is established. Conversely, on the symmetric graded time mesh, through the introduction of a discrete fractional Gronwall inequality, the stability and globally optimal time error estimate can be obtained. Finally, some numerical tests are presented to validate the theoretical results.

math.NA↗

arXiv2025

Finite element method with Grünwald-Letnikov type approximation in time for a constant time delay subdiffusion equation

In this work, a subdiffusion equation with constant time delay $τ$ is considered. First, the regularity of the solution to the considered problem is investigated, finding that its first-order time derivative exhibits singularity at $t=0^+$ and its second-order time derivative shows singularity at both $t=0^+$ and $τ^+$, while the solution can be decomposed into its singular and regular components. Then, we derive a fully discrete finite element scheme to solve the considered problem based on the standard Galerkin finite element method in space and the Grünwald-Letnikov type approximation in time. The analysis shows that the developed numerical scheme is stable. In order to discuss the error estimate, a new discrete Gronwall inequality is established. Under the above decomposition of the solution, we obtain a local error estimate in time for the developed numerical scheme. Finally, some numerical tests are provided to support our theoretical analysis.

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arXiv2024

Local convergence analysis of L1/finite element scheme for a constant delay reaction-subdiffusion equation with uniform time mesh

The aim of this paper is to develop a refined error estimate of L1/finite element scheme for a reaction-subdiffusion equation with constant delay $τ$ and uniform time mesh. Under the non-uniform multi-singularity assumption of exact solution in time, the local truncation errors of the L1 scheme with uniform mesh is investigated. Then we introduce a fully discrete finite element scheme of the considered problem. Next, a novel discrete fractional Grönwall inequality with constant delay term is proposed, which does not include the increasing Mittag-Leffler function comparing with some popular other cases. By applying this Grönwall inequality, we obtain the pointwise-in-time and piecewise-in-time error estimates of the finite element scheme without the Mittag-Leffler function. In particular, the latter shows that, for the considered interval $((i-1)τ,iτ]$, although the convergence in time is low for $i=1$, it will be improved as the increasing $i$, which is consistent with the factual assumption that the smoothness of the solution will be improved as the increasing $i$. Finally, we present some numerical tests to verify the developed theory.

math.NA↗

arXiv2021

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler--Maruyama approximation

This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler--Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler--Maruyama method can reach strong first-order superconvergence when $α= 1$. Finally, several numerical tests are reported for verification of the theoretical findings.

math.NA↗

arXiv2018

Uniform convergence of multigrid finite element method for time-dependent Riesz tempered fractional problem

In this article a theoretical framework for the Galerkin finite element approximation to the time-dependent Riesz tempered fractional problem is provided without the fractional regularity assumption. Because the time-dependent problems should become easier to solve as the time step $τ\rightarrow 0$, which correspond to the mass matrix dominant [R. E. Bank and T. Dupont, {\em Math. Comp.}, 153 (1981), pp. 35--51]. Based on the introduced and analysis of the fractional $τ$-norm, the uniform convergence estimates of the V-cycle multigrid method with the time-dependent fractional problem is strictly proved, which means that the convergence rate of the V-cycle MGM is independent of the mesh size $h$ and the time step $τ$. The numerical experiments are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$ complexity by the fast Fourier transform method, where $N$ is the number of the grid points. To the best of our knowledge, this is the first proof for the convergence rate of the V-cycle multigrid finite element method with $τ\rightarrow 0$.

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arXiv2018

Parallel-in-Time with Fully Finite Element Multigrid for 2-D Space-fractional Diffusion Equations

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions. We firstly obtain a fully discrete scheme via using the linear finite element method to discretize spatial and temporal derivatives to propagate solutions. Next, we present a non-intrusive time-parallelization and its two-level convergence analysis, where we algorithmically and theoretically generalize the MGRIT to time-dependent fine time-grid propagators. Finally, numerical illustrations show that the obtained numerical scheme possesses the saturation error order, theoretical results of the two-level variant deliver good predictions, and significant speedups can be achieved when compared to parareal and the sequential time-stepping approach.

math.NA↗

arXiv2017

Time-space Finite Element Adaptive AMG for Multi-term Time Fractional Advection Diffusion Equations

In this study we construct a time-space finite element (FE) scheme and furnish cost-efficient approximations for one-dimensional multi-term time fractional advection diffusion equations on a bounded domain $Ω$. Firstly, a fully discrete scheme is obtained by the linear FE method in both temporal and spatial directions, and many characterizations on the resulting matrix are established. Secondly, the condition number estimation is proved, an adaptive algebraic multigrid (AMG) method is further developed to lessen computational cost and analyzed in the classical framework. Finally, some numerical experiments are implemented to reach the saturation error order in the $L^2(Ω)$ norm sense, and present theoretical confirmations and predictable behaviors of the proposed algorithm.

math.NA↗

arXiv2017

Fully Finite Element Adaptive Algebraic Multigrid Method for Time-Space Caputo-Riesz Fractional Diffusion Equations

The paper aims to establish a fully discrete finite element (FE) scheme and provide cost-effective solutions for one-dimensional time-space Caputo-Riesz fractional diffusion equations on a bounded domain $Ω$. Firstly, we construct a fully discrete scheme of the linear FE method in both temporal and spatial directions, derive many characterizations on the coefficient matrix and numerically verify that the fully FE approximation possesses the saturation error order under $L^2(Ω)$ norm. Secondly, we theoretically prove the estimation $1+\mathcal{O}(τ^αh^{-2β})$ on the condition number of the coefficient matrix, in which $τ$ and $h$ respectively denote time and space step sizes. Finally, on the grounds of the estimation and fast Fourier transform, we develop and analyze an adaptive algebraic multigrid (AMG) method with low algorithmic complexity, reveal a reference formula to measure the strength-of-connection tolerance which severely affect the robustness of AMG methods in handling fractional diffusion equations, and illustrate the well robustness and high efficiency of the proposed algorithm compared with the classical AMG, conjugate gradient and Jacobi iterative methods.

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