Two fast finite difference methods for a class of variable-coefficient fractional diffusion equations with time delay

IF 3.4 2区数学 Q1 MATHEMATICS, APPLIED

Communications in Nonlinear Science and Numerical Simulation Pub Date : 2024-09-21 DOI:10.1016/j.cnsns.2024.108358

Xue Zhang , Xian-Ming Gu , Yong-Liang Zhao

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引用次数: 0

Abstract

This paper introduces the fast Crank–Nicolson (CN) and compact difference schemes for solving the one- and two-dimensional fractional diffusion equations with time delay. The CN method is employed for temporal discretization, while the fractional centered difference (FCD) formula discretizes the Riesz space derivative. Additionally, a novel fourth-order scheme is developed using compact difference operators to improve accuracy. The convergence and stability of these schemes are rigorously proven. The discretized systems combining Toeplitz-like structures can be effectively solved by Krylov subspace solvers with suitable preconditioners. Each time level of these methods requires a computational complexity of

O (p log p)

per iteration and a memory of

O (p)

, where

p

represents the total number of grid points in space. Numerical examples are given to illustrate both the theoretical results and the computational efficiency of the fast algorithm.

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一类有时间延迟的可变系数分数扩散方程的两种快速有限差分法

本文介绍了用于求解有时间延迟的一维和二维分数扩散方程的快速 Crank-Nicolson（CN）和紧凑差分方案。CN 方法用于时间离散化，而分数中心差分 (FCD) 公式则用于离散化 Riesz 空间导数。此外，还利用紧凑差分算子开发了一种新的四阶方案，以提高精度。这些方案的收敛性和稳定性得到了严格证明。结合了类似托普利兹结构的离散化系统可以通过克雷洛夫子空间求解器和合适的预处理器有效求解。这些方法的每个时间级每次迭代所需的计算复杂度为 O(plogp)，内存为 O(p)，其中 p 代表空间中网格点的总数。本文给出了数值示例，以说明快速算法的理论结果和计算效率。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.