×
Tang, Shi-Ping; Huang, Yu-Mei

A matrix splitting preconditioning method for solving the discretized tempered fractional diffusion equations. (English) Zbl 1508.65021

Numer. Algorithms 92, No. 2, 1311-1333 (2023).
Summary: The initial boundary value problem of the tempered fractional diffusion equations is a kind of important equations arising in many application fields. In this paper, the Crank-Nicolson scheme is applied in the discretization of the tempered fractional diffusion equations. We then get the discretized system of linear equations with the coefficient matrix having the structure of the sum of an identity matrix and the product of a diagonal and a symmetric positive-definite Toeplitz matrix. A scaled diagonal and Toeplitz-approximate splitting (SDTAS) preconditioner is developed, and the GMRES method combined with this preconditioner is applied to solve the linear system. The spectral distribution of the preconditioned matrix is analyzed and some theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method.

Cited in 2 Documents

MSC:

65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
15B05 Toeplitz, Cauchy, and related matrices

Keywords:

tempered fractional diffusion equations; Toeplitz matrix; matrix splitting iteration; preconditioned Krylov subspace iteration method; spectral analysis

Software:

TFPDE
Cite Review PDF
Full Text: DOI

References:

[1] Qu, W.; Liang, Y., Stability and convergence of the Crank-Nicolson scheme for a class of variable-coefficient tempered fractional diffusion equations, Adv. Differ. Equ., 108, 2017 (2017) · Zbl 1422.65180
[2] Li, C.; Deng, W-H, High order schemes for the tempered fractional diffusion equations, Adv. Comput. Math., 42, 543-572 (2016) · Zbl 1347.65136 · doi:10.1007/s10444-015-9434-z
[3] Cartea, A.; Del-Castillo-Negrete, D., Fractional diffusion models of option prices in markets with jumps, Physica. A., 374, 749-763 (2007) · doi:10.1016/j.physa.2006.08.071
[4] He, J-Q; Dong, Y.; Li, S-T; Liu, H-L; Yu, Y-J; Jin, G-Y; Liu, L-D, Study on force distribution of the tempered glass based on laser interference technology, Optik., 126, 5276-5279 (2015) · doi:10.1016/j.ijleo.2015.09.236
[5] Magin, RL, Fractional calculus in bioengineering, Crit. R.v. Biomed Eng., 32, 1-104 (2007)
[6] Rosenau, P., Tempered diffusion: a transport process with propagating fronts and inertial delay, Phys. Rev. A., 46, 7371-7374 (1992) · doi:10.1103/PhysRevA.46.R7371
[7] Chakrabarty, A.; Meerschaert, MM, Tempered stable laws as random walk limits, Stat. Probabil. Lett., 81, 989-997 (2011) · Zbl 1225.60028 · doi:10.1016/j.spl.2011.01.019
[8] Zheng, M.; Karniadakis, GE, Numerical methods for SPDEs with tempered stable processes, SIAM J. Sci. Comput., 37, A1197-A1217 (2015) · Zbl 1320.65020 · doi:10.1137/140966083
[9] Chan, R., Jin, X.-Q.: An introduction to iterative Toeplitz solvers. SIAM, Philadelphia, PA (2007) · Zbl 1146.65028
[10] Saad, Y.: Iterative Methods for Sparse Linear Systems. 2nd Edn. SIAM, Philadelphia, PA (2003) · Zbl 1031.65046
[11] Bai, Z-Z, Motivations and realizations of Krylov subspace methods for large sparse linear systems, J. Comput. Appl. Math., 283, 71-78 (2015) · Zbl 1311.65032 · doi:10.1016/j.cam.2015.01.025
[12] Bai, Z-Z; Golub, GH; Ng, MK, Hermitian and skew-hermitian splitting methods for non-hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24, 603-626 (2003) · Zbl 1036.65032 · doi:10.1137/S0895479801395458
[13] Bai, Z.-Z., Pan, J.-Y.: Matrix analysis and computations. SIAM, Philadelphia PA (2021) · Zbl 07417710
[14] Pan, J-Y; Ke, R-H; Ng, MK; Sun, H-W, Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations, SIAM J. Sci. Comput., 36, A2698-A2719 (2014) · Zbl 1314.65112 · doi:10.1137/130931795
[15] Donatelli, D.; Mazza, M.; Serra-Capizzano, S., Spectral analysis and structure preserving preconditioners for fractional diffusion equations, J. Comput. Phy., 307, 262-279 (2016) · Zbl 1352.65305 · doi:10.1016/j.jcp.2015.11.061
[16] Lin, F-R; Yang, S-W; Jin, X-Q, Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256, 109-117 (2014) · Zbl 1349.65314 · doi:10.1016/j.jcp.2013.07.040
[17] Moroney, T.; Yang, Q-Q, A banded preconditioner for the two-sided nonlinear space-fractional diffusion equation, Comput. Math. Appl., 66, 659-667 (2013) · Zbl 1346.35219 · doi:10.1016/j.camwa.2013.01.048
[18] Lei, S-L; Sun, H-W, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242, 715-725 (2013) · Zbl 1297.65095 · doi:10.1016/j.jcp.2013.02.025
[19] Lu, K-Y, Diagonal and circulant or skew-circulant splitting preconditioners for spatial fractional diffusion equations, Comput. Appl. Math., 37, 4196-4218 (2018) · Zbl 1402.65018 · doi:10.1007/s40314-017-0570-6
[20] Bai, Z-Z; Lu, K-Y; Pan, J-Y, Diagonal and Toeplitz splitting iteration methods for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations, Numer. Linear Algebra Appl., 24, e2093, 1-15 (2017) · Zbl 1463.65040
[21] Lu, K-Y; Xie, D-X; Chen, F.; Muratova, GV, Dominant Hermitian splitting iteration method for discrete space-fractional diffusion equations, Appl. Numer. Math., 164, 15-28 (2021) · Zbl 1460.65035 · doi:10.1016/j.apnum.2020.03.005
[22] Lin, X-L; Ng, MK; Sun, H-W, A splitting preconditioner for Toeplitz-like linear systems arising from fractional diffusion equations, SIAM J. Matrix Anal. Appl., 38, 1580-1614 (2017) · Zbl 1381.65030 · doi:10.1137/17M1115447
[23] Lin, X-L; Ng, MK; Sun, H-W, Efficient preconditioner of one-sided space fractional diffusion equation, BIT Numer. Math., 58, 729-748 (2018) · Zbl 1404.65136 · doi:10.1007/s10543-018-0699-8
[24] She, Z-H; Lao, C-X; Yang, H.; Lin, F-R, Banded preconditioners for Riesz space fractional diffusion equations, J. Sci. Comput., 86, 1-22 (2021) · Zbl 1475.65079 · doi:10.1007/s10915-020-01398-4
[25] Bai, Z-Z; Lu, K-Y, Optimal rotated block-diagonal preconditioning for discretized optimal control problems constrained with fractional time-dependent diffusive equations, Appl. Numer. Math., 163, 126-146 (2021) · Zbl 1466.49029 · doi:10.1016/j.apnum.2021.01.011
[26] Tang, S-P; Huang, Y-M, A DRCS preconditioning iterative method for a constrained fractional optimal control problem, Comput. Appl. Math., 40, 266 (2021) · Zbl 1476.65047 · doi:10.1007/s40314-021-01654-9
[27] Bini, D., Benedetto, F.D.: A new preconditioner for the parallel solution of positive definite Toeplitz systems. ACM Symposium on Parallel Algorithms & Architectures, pp. 220-223 (1990)
[28] Lu, X.; Fang, Z-W; Sun, H-W, Splitting preconditioning based on sine transform for time-dependent Riesz space fractional diffusion equations, J. Appl. Math. Comput., 66, 673-700 (2021) · Zbl 1475.65015 · doi:10.1007/s12190-020-01454-0
[29] Axelsson, O., Iterative solution methods (1994), Cambridge: Cambridge University Press, Cambridge · Zbl 0795.65014 · doi:10.1017/CBO9780511624100
[30] Varga, RS, Geršgorin and his circles (2004), Berlin: Springer, Berlin · Zbl 1057.15023 · doi:10.1007/978-3-642-17798-9
[31] Zygmund, A., Trigonometric series, vol. 21, 6498-6498 (1959), Cambridge: Cambridge University Press, Cambridge · Zbl 0085.05601
[32] Serra-Capizzano, S., Superlinear PCG methods for symmetric Toeplitz systems, Math. Comput., 68, 793-804 (1999) · Zbl 1043.65066 · doi:10.1090/S0025-5718-99-01045-5
[33] Serra-Capizzano, S., Generalized locally Toeplitz sequences: spectral analysis and applications to discretized differential equations, Linear Algebra Appl., 366, 371-402 (2003) · Zbl 1028.65109 · doi:10.1016/S0024-3795(02)00504-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.