Communications on Applied Mathematics and Computation >

2020 , Vol. 2 >Issue 2: 215 - 239

DOI: https://doi.org/10.1007/s42967-019-00050-9

An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients

Expand
  • 1 Department of Mathematics, Hong Kong Baptist University, Hong Kong, China;
    2 School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China;
    3 Department of Mathematics, The University of Hong Kong, Hong Kong, China;
    4 Department of Mathematics, University of Macau, Macao, China

Received date: 2019-06-02

  Revised date: 2019-10-11

  Online published: 2020-02-19

Supported by

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.

Fold

Abstract

In this paper, a second-order fnite-diference scheme is investigated for time-dependent space fractional difusion equations with variable coefcients. 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 coefcients. 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 efciency of the proposed scheme.

Key words： One-side space fractional difusion equation; Variable difusion coefcients; Stability and convergence; High-order fnite-diference scheme; Preconditioner

Cite this article

Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong . An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients[J]. Communications on Applied Mathematics and Computation, 2020 , 2(2) : 215 -239 . DOI: 10.1007/s42967-019-00050-9

References

1. Abirami, A., Prakash, P., Thangavel, K.:Fractional difusion equation-based image denoising model using CN-GL scheme. Int. J. Comput. Math. 95(6/7), 1222-1239 (2018)
2. Chen, M.H., Deng, W.H.:Fourth order accurate scheme for the space fractional difusion equations. SIAM J. Numer. Anal. 52, 1418-1438 (2014)
3. Chen, Y., Vinagre, B.M.:A new IIR-type digital fractional order diferentiator. Signal Process. 83(11), 2359-2365 (2003)
4. Ciarlet, P.G.:Linear and Nonlinear Functional Analysis with Applications, vol. 130. SIAM, Philadelphia (2013)
5. Ding, H., Li, C.:A high-order algorithm for time-Caputo-tempered partial diferential equation with Riesz derivatives in two spatial dimensions. J. Sci. Comput. 80(1), 81-109 (2019)
6. Donatelli, M., Mazza, M., Serra-Capizzano, S.:Spectral analysis and structure preserving preconditioners for fractional difusion equations. J. Comput. Phys. 307, 262-279 (2016)
7. De Hoog, F.:A new algorithm for solving Toeplitz systems of equations. Linear Algebra Appl. 88, 123-138 (1987)
8. Gohberg, I., Olshevsky, V.:Circulants, displacements and decompositions of matrices. Integr. Equ. Oper. Theory 15, 730-743 (1992)
9. Hao, Z.P., Sun, Z.Z., Cao, W.R.:A fourth-order approximation of fractional derivatives with its applications. J. Comput. Phys. 281, 787-805 (2015)
10. Ilic, M., Liu, F., Turner, I., Anh, V.:Numerical approximation of a fractional-in-space difusion equation (II)-with nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal. 9(4), 333-349 (2006)
11. Jin, X.Q., Vong, S.W.:An Introduction to Applied Matrix Analysis. Higher Education Press, Beijing (2016)
12. Koeller, R.:Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51(2), 299-307 (1984)
13. Laub, A.J.:Matrix Analysis for Scientists and Engineers, vol. 91. SIAM, Philadelphia (2005)
14. Lei, S.L., Chen, X., Zhang, X.H.:Multilevel circulant preconditioner for high-dimensional fractional difusion equations. East Asian J. Appl. Math. 6, 109-130 (2016)
15. Lei, S.L., Huang, Y.C.:Fast algorithms for high-order numerical methods for space-fractional difusion equations. Int. J. Comput. Math. 94(5), 1062-1078 (2017)
16. Lei, S.L., Sun, H.W.:A circulant preconditioner for fractional difusion equations. J. Comput. Phys. 242, 715-725 (2013)
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)
18. Li, M., Gu, X.M., Huang, C., Fei, M., Zhang, G.:A fast linearized conservative fnite element method for the strongly coupled nonlinear fractional Schrödinger equations. J. Comput. Phys. 358, 256-282 (2018)
19. Lin, X.L., Ng, M.K.:A fast solver for multidimensional time-space fractional difusion equation with variable coefcients. Comput. Math. Appl. 78(5), 1477-1489 (2019)
20. Lin, X.L., Ng, M.K., Sun, H.W.:Efcient preconditioner of one-sided space fractional difusion equation. BIT 58, 729-748 (2018). https://doi.org/10.1007/s10543-018-0699-8
21. Lin, X.L., Ng, M.K., Sun, H.W.:Stability and convergence analysis of fnite diference schemes for time-dependent space-fractional difusion equations with variable difusion coefcients. J. Sci. Comput. 75, 1102-1127 (2018)
22. Lin, X.L., Ng, M.K., Sun, H.W.:Crank-Nicolson alternative direction implicit method for spacefractional difusion equations with nonseparable coefcients. SIAM J. Numer. Anal. 57(3), 997-1019 (2019)
23. Meerschaert, M.M., Schefer, H.P., Tadjeran, C.:Finite diference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211, 249-261 (2006)
24. Meerschaert, M.M., Tadjeran, C.:Finite diference approximations for fractional advection-dispersion fow equations. J. Comput. Appl. Math. 172(1), 65-77 (2004)
25. Moghaddam, B., Machado, J.T., Morgado, M.:Numerical approach for a class of distributed order time fractional partial diferential equations. Appl. Numer. Math. 136, 152-162 (2019)
26. Moghaderi, H., Dehghan, M., Donatelli, M., Mazza, M.:Spectral analysis and multigrid preconditioners for two-dimensional space-fractional difusion equations. J. Comput. Phys. 350, 992-1011 (2017)
27. Ng, M.K.:Iterative Methods for Toeplitz Systems. Oxford University Press, USA (2004)
28. Osman, S., Langlands, T.:An implicit Keller Box numerical scheme for the solution of fractional subdifusion equations. Appl. Math. Comput. 348, 609-626 (2019)
29. Podlubny, I.:Fractional Diferential Equations. Academic Press, New York (1999)
30. Pu, Y.F., Siarry, P., Zhou, J.L., Zhang, N.:A fractional partial diferential equation based multiscale denoising model for texture image. Math. Methods Appl. Sci. 37(12), 1784-1806 (2014)
31. Pu, Y.F., Zhou, J.L., Yuan, X.:Fractional diferential mask:a fractional diferential-based approach for multiscale texture enhancement. IEEE Trans. Image Process. 19(2), 491-511 (2009)
32. Pu, Y.F., Zhou, J.L., 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, S.L., Vong, S.:A note on the stability of a second order fnite diference scheme for space fractional difusion equations. Numer. Algebra Control Optim. 4, 317-325 (2014)
34. Rossikhin, Y.A., Shitikova, M.V.: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, S.G., Kilbas, A.A., Marichev, O.I.:Fractional Integrals and Derivatives:Theory and Applications. Gordon and Breach Science Publishers, Switzerland (1993)
37. Sousa, E.:Numerical approximations for fractional difusion equations via splines. Comput. Math. Appl. 62(3), 938-944 (2011)
38. Sousa, E., Li, C.:A weighted fnite diference method for the fractional difusion equation based on the Riemann-Liouville derivative. Appl. Numer. Math. 90, 22-37 (2015)
39. Tadjeran, C., Meerschaert, M.M.:A second-order accurate numerical method for the two-dimensional fractional difusion equation. J. Comput. Phys. 220, 813-823 (2007)
40. Tadjeran, C., Meerschaert, M.M., Schefer, H.P.:A second-order accurate numerical approximation for the fractional difusion equation. J. Comput. Phys. 213, 205-213 (2006)
41. Tian, W.Y., Zhou, H., Deng, W.H.:A class of second order diference approximations for solving space fractional difusion equations. Math. Comp. 84, 1703-1727 (2015)
42. Tseng, C.C.:Design of fractional order digital FIR diferentiators. IEEE Signal Process Lett. 8(3), 77-79 (2001)
43. Vong, S., Lyu, P.:On a second order scheme for space fractional difusion equations with variable coefcients. Appl. Numer. Math. 137, 34-48 (2019)
44. Vong, S., Lyu, P., Chen, X., Lei, S.L.:High order fnite diference method for time-space fractional diferential equations with Caputo and Riemann-Liouville derivatives. Numer. Algorithms 72, 195- 210 (2016)
45. Yuttanan, B., Razzaghi, M.:Legendre wavelets approach for numerical solutions of distributed order fractional diferential equations. Appl. Math. Modell. 70, 350-364 (2019)
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-difusion equation. SIAM J. Numer. Anal. 52, 2599-2622 (2014)
47. Zhuang, P., Liu, F., Anh, V., Turner, I.:Numerical methods for the variable-order fractional advection-difusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47(3), 1760-1781 (2009)
Options
Outlines

/

〈 〉

Copyright © Editorial By Communications on Applied Mathematics and Computation
沪ICP备09014157