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26.03.2020 | Original Paper

# A second-order fast compact scheme with unequal time-steps for subdiffusion problems

verfasst von: Xin Li, Hong-lin Liao, Luming Zhang

Erschienen in: Numerical Algorithms | Ausgabe 3/2021

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## Abstract

In consideration of the initial singularity of the solution, a temporally second-order fast compact difference scheme with unequal time-steps is presented and analyzed for simulating the subdiffusion problems in several spatial dimensions. On the basis of sum-of-exponentials technique, a fast Alikhanov formula is derived on general nonuniform meshes to approximate the Caputo’s time derivative. Meanwhile, the spatial derivatives are approximated by the fourth-order compact difference operator, which can be implemented by a fast discrete sine transform via the FFT algorithm. So the proposed algorithm is computationally efficient with the computational cost about $$O(MN\log M\log N)$$ and the storage requirement $$O(M\log N)$$, where M and N are the total numbers of grids in space and time, respectively. With the aids of discrete fractional Grönwall inequality and global consistency analysis, the unconditional stability and sharp H1-norm error estimate reflecting the regularity of solution are established rigorously by the discrete energy approach. Three numerical experiments are included to confirm the sharpness of our analysis and the effectiveness of our fast algorithm.
Literatur
1.
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) CrossRef
2.
Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)
3.
Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
4.
Li, C.P., Chen, A., Ye, J.J.: Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230, 3352–3368 (2011)
5.
Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)
6.
Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
7.
Lv, C.W., Xu, C.J.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38, A2699–A2724 (2016)
8.
Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D frational subdiffusion problems. J. Comput. Phys. 229, 6613–6622 (2010)
9.
McLean, W.: Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52, 123–138 (2010)
10.
Jin, B.T., Lazarov, R., Zhou, Z.: An analysis of the l1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36(1), 197–221 (2016)
11.
Sakamoto, K., Yamamoto, M.: Initial value/boundary value prolems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)
12.
Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)
13.
Yuste, S.B., Quintana-Murillo, J.: A finite difference method with non-uniform timesteps for fractional diffusion equation. Comput. Phys. Commun. 183, 2594–2600 (2012)
14.
Mustapha, K., Aimutawa, J.: A finite difference method for an anomalous sub-diffusion equation, theory and applications. Numer. Algo. 61, 525–543 (2012)
15.
Zhang, Y.N., Sun, Z.Z., Liao, H.-L.: Finite difference methods for the time fractional diffusion equation on nonuniform meshes. J. Comput. Phys. 265, 195–210 (2014)
16.
Li, C.P., Yi, Q., Chen, A.: Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Appl. Math. 316, 614–631 (2016)
17.
Liao, H.-L., Li, D.F., Zhang, J.W.: Sharp error estimate of the nonuniform l1 formula for linear reaction-subdiffusion equation. SIAM J. Numer. Anal. 56, 1112–1133 (2018)
18.
Liao, H.-L., McLean, W., Zhang, J.W.: A discrete grönwall inequality with application to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57, 218–237 (2019)
19.
Liao, H.-L., McLean, W., Zhang, J.W.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem. arXiv: 1803.​09873v2 (2018)
20.
Ren, J.C., Liao, H.-L., Zhang, J.W., Zhang, Z.M.: Sharp h 1-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems. arXiv: 1811.​08059v1 (2018)
21.
Ke, R.H., Ng, M.K., Sun, H.W.: A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equation. J. Comput. Phys. 303, 203–211 (2015)
22.
Baffet, D., Hesthaven, J.S.: A kernel compression scheme for fractional differential equations. SIAM J. Numer. Anal. 55, 496–520 (2017)
23.
Jiang, S.D., Zhang, J.W., Zhang, Q., Zhang, Z.M.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21, 650–678 (2017)
24.
Yan, Y.G., Sun, Z.Z., Zhang, J.W.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme. Commun. Comput. Phys. 22, 1028–1048 (2017)
25.
Shen, J.Y., Sun, Z.Z., Du, R.: Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time. East Asia J. Appl. Math. 8, 834–858 (2018)
26.
Liao, H.-L., Yan, Y.G., Zhang, J.W.: Unconditional convergence of a fast two-level linearized algorithm for semilinear subdiffusion equations. J. Sci. Comput. 80, 1–25 (2019)
27.
Wang, H.Q., Zhang, Y., Ma, X., Qiu, J., Liang, Y.: An efficient implementation of fourth-order compact finite difference scheme for Poisson equation with Dirichlet boundary conditions. Commun. Math. Appl. 71, 1843–1860 (2016)
28.
Wang, H.Q., Ma, X., Lu, J.L., Cao, W.: An efficient time-splitting compact finite difference method for Gross-Pitaevskii equation. Appl. Math. Comput. 297, 131–144 (2017)
29.
Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011) CrossRef
Titel
A second-order fast compact scheme with unequal time-steps for subdiffusion problems
verfasst von
Xin Li
Hong-lin Liao
Luming Zhang
Publikationsdatum
26.03.2020
Verlag
Springer US
Erschienen in
Numerical Algorithms / Ausgabe 3/2021
Print ISSN: 1017-1398
Elektronische ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-020-00920-x

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