Efficient ADI schemes and preconditioning for a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients. (English) Zbl 1505.65284
Summary: In this paper, alternating direction implicit (ADI) finite difference method and preconditioned Krylov subspace method are combined to solve a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients. We prove the unconditional stability and convergence rate of the ADI finite difference method provided that the diffusion coefficients satisfy the given conditions. For the linear system in each spatial direction, we establish a circulant approximate inverse preconditioner to accelerate the Krylov subspace method. In addition, we also use matrix-free algorithms and fast Fourier transforms (FFT) to speed up the solution of linear systems. Numerical experiments show the utility of the ADI method and the preconditioner.
MSC:
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
| 65F10 | Iterative numerical methods for linear systems |
| 65F55 | Numerical methods for low-rank matrix approximation; matrix compression |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 15B05 | Toeplitz, Cauchy, and related matrices |
| 65T50 | Numerical methods for discrete and fast Fourier transforms |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
Keywords:
high-dimensional fractional diffusion equations; alternating direction implicit method; Toeplitz matrix; approximate inverse preconditioner; fast Fourier transform; Krylov subspace methodSoftware:
ma2dfcReferences:
| [1] | Hilfer, R., Applications of Fractional Calculus in Physics (2000), World scientific · Zbl 0998.26002 |
| [2] | Kumar, P.; Chaudhary, S. K., Analysis of fractional order control system with performance and stability, Int. J. Eng. Sci. Tech., 9, 408-416 (2017) |
| [3] | Kirchner, J. W.; Feng, X.; Neal, C., Fractal stream chemistry and its implications for contaminant transport in catchments, Nature, 403, 6769, 524-527 (2000) |
| [4] | Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., Application of a fractional advection-dispersion equation, Water Resour. Res., 36, 6, 1403-1412 (2000) |
| [5] | Carreras, B.; Lynch, V.; Zaslavsky, G., Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8, 12, 5096-5103 (2001) |
| [6] | Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339, 1, 1-77 (2000) · Zbl 0984.82032 |
| [7] | Xu, Y.; Sun, H.-W.; Sheng, Q., On variational properties of balanced central fractional derivatives, Int. J. Comput. Math., 95, 6-7, 1195-1209 (2018) · Zbl 1513.65039 |
| [8] | Mao, Z.; Shen, J., Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients, J. Comput. Phys., 307, 243-261 (2016) · Zbl 1352.65395 |
| [9] | Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010 |
| [10] | Tadjeran, C.; Meerschaert, M. M.; Scheffler, H.-P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213, 1, 205-213 (2006) · Zbl 1089.65089 |
| [11] | Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 2, 1533-1552 (2007) · Zbl 1126.65121 |
| [12] | Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, Y.; Jara, B. M.V., Matrix approach to discrete fractional calculus II: Partial fractional differential equations, J. Comput. Phys., 228, 8, 3137-3153 (2009) · Zbl 1160.65308 |
| [13] | Podlubny, I., Matrix approach to discrete fractional calculus, Fract. Calc. Appl. Anal., 3, 4, 359-386 (2000) · Zbl 1030.26011 |
| [14] | Roop, J. P., Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R \({}^2\), J. Comput. Appl. Math., 193, 1, 243-268 (2006) · Zbl 1092.65122 |
| [15] | Meerschaert, M. M.; Scheffler, H.-P.; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211, 1, 249-261 (2006) · Zbl 1085.65080 |
| [16] | Wang, H.; Du, N., Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations, J. Comput. Phys., 258, 305-318 (2014) · Zbl 1349.65342 |
| [17] | Lei, S.-L.; Fan, D.; Chen, X., Fast solution algorithms for exponentially tempered fractional diffusion equations, Numer. Methods Part. Differ. Equ., 34, 4, 1301-1323 (2018) · Zbl 1407.65112 |
| [18] | Lin, X.-L.; Ng, M. K.; Sun, H.-W., Stability and convergence analysis of finite difference schemes for time-dependent space-fractional diffusion equations with variable diffusion coefficients, J. Sci. Comput., 75, 2, 1102-1127 (2018) · Zbl 1398.65214 |
| [19] | Lin, X.-L.; Ng, M. K.; Sun, H.-W., Crank-Nicolson alternative direction implicit method for space-fractional diffusion equations with nonseparable coefficients, SIAM J. Numer. Anal., 57, 3, 997-1019 (2019) · Zbl 1422.65162 |
| [20] | Sun, Z.-Z.; Gao, G.-H., Fractional differential equations, (Fractional Differential Equations (2020), De Gruyter) · Zbl 1440.65003 |
| [21] | Li, C.; Cai, M., Theory and Numerical Approximations of Fractional Integrals and Derivatives (2019), SIAM |
| [22] | Chou, L.-K.; Lei, S.-L., Fast ADI method for high dimensional fractional diffusion equations in conservative form with preconditioned strategy, Comput. Math. Appl., 73, 3, 385-403 (2017) · Zbl 1368.65165 |
| [23] | Klimek, M.; Agrawal, O. P., Fractional Sturm-Liouville problem, Comput. Math. Appl., 66, 5, 795-812 (2013) · Zbl 1348.34018 |
| [24] | Zayernouri, M.; Karniadakis, G. E., Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, J. Comput. Phys., 252, 495-517 (2013) · Zbl 1349.34095 |
| [25] | Klimek, M.; Blasik, M., Regular fractional Sturm-Liouville problem with discrete spectrum: Solutions and applications, (ICFDA’14 International Conference on Fractional Differentiation and Its Applications 2014 (2014)), 1-6 |
| [26] | Derakhshan, M. H.; Ansari, A., Numerical approximation to Prabhakar fractional Sturm-Liouville problem, J. Comput. Appl. Math., 38, 2, 1-20 (2019) · Zbl 1449.65163 |
| [27] | Fang, Z.-W.; Ng, M. K.; Sun, H.-W., Circulant preconditioners for a kind of spatial fractional diffusion equations, Numer. Algorithms, 82, 2, 729-747 (2019) · Zbl 1437.65013 |
| [28] | Pang, H.-K.; Qin, H.-H.; Sun, H.-W.; Ma, T.-T., Circulant-based approximate inverse preconditioners for a class of fractional diffusion equations, Comput. Math. Appl., 85, 18-29 (2021) · Zbl 1524.65124 |
| [29] | Pan, J.; Ke, R.; Ng, M. K.; Sun, H.-W., Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations, SIAM J. Sci. Comput., 36, 6, A2698-A2719 (2014) · Zbl 1314.65112 |
| [30] | Brigham, E. O., The Fast Fourier Transform and Its Applications (1988), Prentice-Hall, Inc. |
| [31] | Zhang, Y.-N.; Sun, Z.-N., Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230, 24, 8713-8728 (2011) · Zbl 1242.65174 |
| [32] | Strang, G., A proposal for Toeplitz matrix calculations, Stud. Appl. Math., 74, 2, 171-176 (1986) · Zbl 0621.65025 |
| [33] | Chan, R. H.-F.; Jin, X.-Q., An Introduction to Iterative Toeplitz Solvers (2007), SIAM · Zbl 1146.65028 |
| [34] | Strohmer, T., Four short stories about Toeplitz matrix calculations, Linear Algebra Appl., 343, 321-344 (2002) · Zbl 0999.65026 |
| [35] | Varah, J. M., A lower bound for the smallest singular value of a matrix, Linear Algebra Appl., 11, 1, 3-5 (1975) · Zbl 0312.65028 |
| [36] | Chan, R. H.; Strang, G., Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Stat. Comput., 10, 1, 104-119 (1989) · Zbl 0666.65030 |
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.
