A fast implicit difference scheme for solving the generalized time-space fractional diffusion equations with variable coefficients

Autor: Gu, Xian-Ming, Huang, Ting-Zhu, Zhao, Yong-Liang, Lyu, Pin, Carpentieri, Bruno
Rok vydání: 2019
Předmět:
Zdroj: Numerical Methods for Partial Differential Equations, 37(2) (2021):1136-1162
Druh dokumentu: Working Paper
DOI: 10.1002/num.22571
Popis: In this paper, we first propose an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the $L1$-type formula for the generalized Caputo fractional derivative in time discretization and the second-order weighted and shifted Gr\"{u}nwald difference (WSGD) formula in spatial discretization, respectively. Theoretical results and numerical tests are conducted to verify the $(2 - \gamma)$-order and 2-order of temporal and spatial convergence with $\gamma\in(0,1)$ the order of Caputo fractional derivative, respectively. The fast sum-of-exponential approximation of the generalized Caputo fractional derivative and Toeplitz-like coefficient matrices are also developed to accelerate the proposed implicit difference scheme. Numerical experiments show the effectiveness of the proposed numerical scheme and its good potential for large-scale simulation of GTSFDEs.
Comment: 23 pages, 10 tables, 1 figure. Make several corrections again and have been submitted to a journal at Sept. 20, 2019. Version 2: Make some necessary corrections and symbols, 13 Jan. 2020. Revised manuscript has been resubmitted to journal
Databáze: arXiv
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