×

An efficient second-order convergent scheme for one-side space fractional diffusion equations with variable coefficients. (English) Zbl 1463.65233

Summary: 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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
15B05 Toeplitz, Cauchy, and related matrices
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abirami, A.; Prakash, P.; Thangavel, K., Fractional diffusion equation-based image denoising model using CN-GL scheme, Int. J. Comput. Math., 95, 6-7, 1222-1239 (2018) · Zbl 1496.94005
[2] Chen, Mh; Deng, Wh, Fourth order accurate scheme for the space fractional diffusion equations, SIAM J. Numer. Anal., 52, 1418-1438 (2014) · Zbl 1318.65048
[3] Chen, Y.; Vinagre, Bm, A new IIR-type digital fractional order differentiator, Signal Process., 83, 11, 2359-2365 (2003) · Zbl 1145.93423
[4] Ciarlet, Pg, Linear and Nonlinear Functional Analysis with Applications (2013), Philadelphia: SIAM, Philadelphia · Zbl 1293.46001
[5] Ding, H.; Li, C., A high-order algorithm for time-Caputo-tempered partial differential equation with Riesz derivatives in two spatial dimensions, J. Sci. Comput., 80, 1, 81-109 (2019) · Zbl 1447.35351
[6] Donatelli, M.; Mazza, M.; Serra-Capizzano, S., Spectral analysis and structure preserving preconditioners for fractional diffusion equations, J. Comput. Phys., 307, 262-279 (2016) · Zbl 1352.65305
[7] De Hoog, F., A new algorithm for solving Toeplitz systems of equations, Linear Algebra Appl., 88, 123-138 (1987) · Zbl 0621.65014
[8] Gohberg, I.; Olshevsky, V., Circulants, displacements and decompositions of matrices, Integr. Equ. Oper. Theory, 15, 730-743 (1992) · Zbl 0772.15010
[9] Hao, Zp; Sun, Zz; Cao, Wr, A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys., 281, 787-805 (2015) · Zbl 1352.65238
[10] Ilic, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation (II)-with nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9, 4, 333-349 (2006) · Zbl 1132.35507
[11] Jin, Xq; Vong, Sw, An Introduction to Applied Matrix Analysis (2016), Beijing: Higher Education Press, Beijing · Zbl 1343.65027
[12] Koeller, R., Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51, 2, 299-307 (1984) · Zbl 0544.73052
[13] Laub, Aj, Matrix Analysis for Scientists and Engineers (2005), Philadelphia: SIAM, Philadelphia · Zbl 1077.15001
[14] Lei, Sl; Chen, X.; Zhang, Xh, Multilevel circulant preconditioner for high-dimensional fractional diffusion equations, East Asian J. Appl. Math., 6, 109-130 (2016) · Zbl 1481.65140
[15] Lei, Sl; Huang, Yc, Fast algorithms for high-order numerical methods for space-fractional diffusion equations, Int. J. Comput. Math., 94, 5, 1062-1078 (2017) · Zbl 1378.65160
[16] Lei, Sl; Sun, Hw, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242, 715-725 (2013) · Zbl 1297.65095
[17] Li, C.; Deng, W.; Zhao, L., Well-posedness and numerical algorithm for the tempered fractional differential equations, Discrete Contin. Dyn. Syst., 24, 4, 1989-2015 (2019) · Zbl 1414.34005
[18] Li, M.; Gu, Xm; Huang, C.; Fei, M.; Zhang, G., A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys., 358, 256-282 (2018) · Zbl 1382.65320
[19] Lin, Xl; Ng, Mk, A fast solver for multidimensional time-space fractional diffusion equation with variable coefficients, Comput. Math. Appl., 78, 5, 1477-1489 (2019) · Zbl 1442.65169
[20] Lin, Xl; Ng, Mk; Sun, Hw, Efficient preconditioner of one-sided space fractional diffusion equation, BIT, 58, 729-748 (2018) · Zbl 1404.65136 · doi:10.1007/s10543-018-0699-8
[21] Lin, Xl; Ng, Mk; Sun, Hw, Stability and convergence analysis of finite difference schemes for time-dependent space-fractional diffusion equations with variable diffusion coefficients, J. Sci. Comput., 75, 1102-1127 (2018) · Zbl 1398.65214
[22] Lin, Xl; Ng, Mk; Sun, Hw, 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
[23] Meerschaert, Mm; Scheffler, Hp; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211, 249-261 (2006) · Zbl 1085.65080
[24] Meerschaert, Mm; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 1, 65-77 (2004) · Zbl 1126.76346
[25] Moghaddam, B.; Machado, Jt; Morgado, M., Numerical approach for a class of distributed order time fractional partial differential equations, Appl. Numer. Math., 136, 152-162 (2019) · Zbl 1407.65122
[26] Moghaderi, H.; Dehghan, M.; Donatelli, M.; Mazza, M., Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations, J. Comput. Phys., 350, 992-1011 (2017) · Zbl 1380.65240
[27] Ng, Mk, Iterative Methods for Toeplitz Systems (2004), USA: Oxford University Press, USA · Zbl 1059.65031
[28] Osman, S.; Langlands, T., An implicit Keller Box numerical scheme for the solution of fractional sub-diffusion equations, Appl. Math. Comput., 348, 609-626 (2019) · Zbl 1429.65195
[29] Podlubny, I., Fractional Differential Equations (1999), New York: Academic Press, New York · Zbl 0918.34010
[30] Pu, Yf; Siarry, P.; Zhou, Jl; Zhang, N., A fractional partial differential equation based multiscale denoising model for texture image, Math. Methods Appl. Sci., 37, 12, 1784-1806 (2014) · Zbl 1301.35203
[31] Pu, Yf; Zhou, Jl; Yuan, X., Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement, IEEE Trans. Image Process., 19, 2, 491-511 (2009) · Zbl 1371.94302
[32] Pu, Yf; Zhou, Jl; Zhang, Y.; Zhang, N.; Huang, G.; Siarry, P., Fractional extreme value adaptive training method: fractional steepest descent approach, IEEE Trans. Neural Networks Learn. Syst., 26, 4, 653-662 (2013)
[33] Qu, W.; Lei, Sl; Vong, S., A note on the stability of a second order finite difference scheme for space fractional diffusion equations, Numer. Algebra Control Optim., 4, 317-325 (2014) · Zbl 1311.65117
[34] Rossikhin, Ya; Shitikova, Mv, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50, 1, 15-67 (1997)
[35] Roy, S., On the realization of a constant-argument immittance or fractional operator, IEEE Trans. Circ. Theory, 14, 3, 264-274 (1967)
[36] Samko, Sg; Kilbas, Aa; Marichev, Oi, Fractional Integrals and Derivatives: Theory and Applications (1993), Switzerland: Gordon and Breach Science Publishers, Switzerland · Zbl 0818.26003
[37] Sousa, E., Numerical approximations for fractional diffusion equations via splines, Comput. Math. Appl., 62, 3, 938-944 (2011) · Zbl 1228.65153
[38] Sousa, E.; Li, C., A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative, Appl. Numer. Math., 90, 22-37 (2015) · Zbl 1326.65111
[39] Tadjeran, C.; Meerschaert, Mm, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys., 220, 813-823 (2007) · Zbl 1113.65124
[40] Tadjeran, C.; Meerschaert, Mm; Scheffler, Hp, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213, 205-213 (2006) · Zbl 1089.65089
[41] Tian, Wy; Zhou, H.; Deng, Wh, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp., 84, 1703-1727 (2015) · Zbl 1318.65058
[42] Tseng, Cc, Design of fractional order digital FIR differentiators, IEEE Signal Process Lett., 8, 3, 77-79 (2001)
[43] Vong, S.; Lyu, P., On a second order scheme for space fractional diffusion equations with variable coefficients, Appl. Numer. Math., 137, 34-48 (2019) · Zbl 1419.65032
[44] Vong, S.; Lyu, P.; Chen, X.; Lei, Sl, High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives, Numer. Algorithms, 72, 195-210 (2016) · Zbl 1382.65259
[45] Yuttanan, B.; Razzaghi, M., Legendre wavelets approach for numerical solutions of distributed order fractional differential equations, Appl. Math. Modell., 70, 350-364 (2019) · Zbl 1466.65054
[46] Zeng, F.; Liu, F.; Li, C.; Burrage, K.; Turner, I.; Anh, V., A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, SIAM J. Numer. Anal., 52, 2599-2622 (2014) · Zbl 1382.65349
[47] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47, 3, 1760-1781 (2009) · Zbl 1204.26013
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.