TY - JOUR
T1 - An Efficient Second-Order Convergent Scheme for One-Side Space Fractional Diffusion Equations with Variable Coefficients
AU - Lin, Xue lei
AU - Lyu, Pin
AU - Ng, Michael K.
AU - Sun, Hai Wei
AU - Vong, Seakweng
N1 - Funding information:
This research was supported by research Grants, 12306616, 12200317, 12300519, 12300218 from HKRGC GRF, 11801479 from NSFC, MYRG2018-00015-FST from University of Macau, and 0118/2018/A3 from FDCT of Macao, Macao Science and Technology Development Fund 0005/2019/A, 050/2017/A, and the Grant MYRG2017-00098-FST and MYRG2018-00047-FST from University of Macau.
Publisher Copyright:
© 2020, Shanghai University.
PY - 2020/6
Y1 - 2020/6
N2 - In this paper, a second-order finite-difference scheme is investigated for time-dependent space fractional diffusion equations with variable coefficients. In the presented scheme, the Crank–Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald–Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the variable coefficients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes, so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efficiency of the proposed scheme.
AB - In this paper, a second-order finite-difference scheme is investigated for time-dependent space fractional diffusion equations with variable coefficients. In the presented scheme, the Crank–Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald–Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the variable coefficients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes, so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efficiency of the proposed scheme.
KW - High-order finite-difference scheme
KW - One-side space fractional diffusion equation
KW - Preconditioner
KW - Stability and convergence
KW - Variable diffusion coefficients
UR - http://www.scopus.com/inward/record.url?scp=85093817748&partnerID=8YFLogxK
U2 - 10.1007/s42967-019-00050-9
DO - 10.1007/s42967-019-00050-9
M3 - Journal article
AN - SCOPUS:85093817748
SN - 2096-6385
VL - 2
SP - 215
EP - 239
JO - Communications on Applied Mathematics and Computation
JF - Communications on Applied Mathematics and Computation
IS - 2
ER -