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A parareal finite volume method for variable-order time-fractional diffusion equations. (English) Zbl 07262120
Summary: In this paper, we investigate the well-posedness and solution regularity of a variable-order time-fractional diffusion equation, which is often used to model the solute transport in complex porous media where the micro-structure of the porous media may changes over time. We show that the variable-order time-fractional diffusion equations have flexible abilities to eliminate the nonphysical singularity of the solutions to their constant-order analogues. We also present a finite volume approximation and provide its stability and convergence analysis in a weighted discrete norm. Furthermore, we develop an efficient parallel-in-time procedure to improve the computational efficiency of the variable-order time-fractional diffusion equations. Numerical experiments are performed to confirm the theoretical results and to demonstrate the effectiveness and efficiency of the parallel-in-time method.
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65Y05 Parallel numerical computation
35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
76S05 Flows in porous media; filtration; seepage
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
[1] Deng, W., Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47, 1, 204-226 (2009) · Zbl 1416.65344
[2] Ervin, V.; Heuer, N.; Roop, J., Regularity of the solution to 1-D fractional order diffusion equations, Math. Comput., 87, 2273-2294 (2018) · Zbl 1394.65145
[3] Evans, LC, Graduate Studies in Mathematics (1998), Rhode Island: American Mathematical Society, Rhode Island
[4] Fu, H.; Liu, H.; Wang, H., A finite volume method for two-dimensional Riemann-Liouville space-fractional diffusion equation and its efficient implementation, J. Comput. Phys., 388, 316-334 (2019)
[5] Fu, H.; Wang, H., A preconditioned fast parareal finite difference method for space-time fractional partial differential equation, J. Sci. Comput., 78, 1724-1743 (2019) · Zbl 1415.65190
[6] Horn, RA; Johnson, CR, Topics in Matrix Analysis (1994), Cambridge: Cambridge University Press, Cambridge
[7] Jia, J.; Zheng, X.; Fu, H.; Dai, P.; Wang, H., A fast method for variable-order space-fractional diffusion equations, Numer. Algorithms (2020)
[8] Jiang, S.; Zhang, J.; Zhang, Q.; Zhang, Z., Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21, 3, 650-678 (2017)
[9] Jiang, W.; Liu, N., A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model, Appl. Numer. Math., 119, 18-32 (2017) · Zbl 1432.65155
[10] Ke, R.; Ng, MK; Sun, HW, A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations., J. Comput. Phys, 303, C, 203-211 (2015) · Zbl 1349.65404
[11] Li, X.; Xu, C., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47, 3, 2108-2131 (2009) · Zbl 1193.35243
[12] Li, C.; Ding, H., Higher order finite difference method for the reaction and anomalous-diffusion equation, Appl. Math. Model., 38, 3802-3821 (2014) · Zbl 1429.65188
[13] Lin, Y.; Xu, C., Finite difference/spectral approximation for the time-fractional diffusion equation, J. Comput. Phys., 225, 1533-1552 (2007) · Zbl 1126.65121
[14] Liu, Y.; Du, Y.; Li, H.; Liu, F.; Wang, Y., Some second-order \(\theta\) schemes combined with finite element method for nonlinear fractional cable equation, Numer. Algorithms, 80, 2, 533-555 (2019) · Zbl 1433.65218
[15] Liu, Z.; Li, X., A Crank-Nicolson difference scheme for the time variable fractional mobile-immobile advection-dispersion equation, J. Appl. Math. Comput., 56, 1-2, 391-410 (2018) · Zbl 1448.65112
[16] Liu, F.; Zhuang, P.; Burrage, K., Numerical methods and analysis for a class of fractional advection-dispersion models, Comput. Math. Appl., 64, 10, 2990-3007 (2012) · Zbl 1268.65124
[17] Ma, H.; Yang, Y., Jacobi spectral collocation method for the time variable-order fractional mobile-immobile advection-dispersion solute transport model, East Asian J. Appl. Math., 6, 3, 337-352 (2016) · Zbl 06797027
[18] Meerschaert M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics (2011) · Zbl 1247.60003
[19] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032
[20] Podlubny, I., Fractional Differential Equations (1999), Cambridge: Academic Press, Cambridge · Zbl 0918.34010
[21] Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 426-447 (2011) · Zbl 1219.35367
[22] Schumer, R.; Benson, DA; Meerschaert, MM; Baeumer, B., Fractal mobile/immobile solute transport, Water Resour. Res., 39, 10, 1296 (2003)
[23] Shao, J., New integral inequalities with weakly singular kernel for discontinuous functions and their applications to impulsive fractional differential systems, J. Appl. Math. (2014) · Zbl 1437.34065
[24] Stynes, M.; O’Riordan, E.; Gracia, JL, Error analysis of a finite difference method on graded mesh for a time-fractional diffusion equation, SIAM Numer. Anal., 55, 1057-1079 (2017) · Zbl 1362.65089
[25] Sun, HG; Chang, A.; Zhang, Y.; Chen, W., A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal., 22, 1, 27-59 (2019) · Zbl 1428.34001
[26] Sun, Z.; Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56, 2, 193-209 (2006) · Zbl 1094.65083
[27] Umarov, SR; Steinberg, ST, Variable order differential equations with piecewise constant order function and diffusion with changing modes, Z. Anal. Anwend., 28, 431-450 (2009) · Zbl 1181.35359
[28] Wang, H.; Zheng, X., Wellposedness and regularity of the variable-order time-fractional diffusion equations, J. Math. Anal. Appl., 475, 1778-1802 (2019) · Zbl 07053183
[29] Wang, H.; Zheng, X., Analysis and numerical solution of a nonlinear variable-order fractional differential equation, Adv. Comput. Math., 45, 2647-2675 (2019)
[30] Wu, S.; Zhou, T., Parareal algorithms with local time-integrators for time fractional differential equations, J. Comput. Phys., 358, 135-149 (2018) · Zbl 1422.65473
[31] Xian, Y.; Jin, M.; Zhan, H.; Liu, Y., Reactive transport of nutrients and bioclogging during dynamic disconnection process of stream and groundwater, Water Resour. Res., 55, 3882-3903 (2019)
[32] Xu, Q.; Hesthaven, JS; Chen, F., A parareal method for time-fractional differential equations, J. Comput. Phys., 293, 173-183 (2015) · Zbl 1349.65220
[33] Yin, B.; Liu, Y.; Li, H.; He, S., Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions, J. Comput. Phys., 379, 351-372 (2019)
[34] Zeng, F.; Zhang, Z.; Karniadakis, GE, A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations, SIAM Sci. Comput., 37, A2710-A2732 (2015) · Zbl 1339.65197
[35] Zhao, X.; Sun, Z.; Karniadakis, GE, Second-order approximations for variable order fractional derivatives: algorithms and applications, J. Comput. Phys., 293, 184-200 (2015) · Zbl 1349.65092
[36] Zhang, H.; Liu, F.; Phanikumar, MS; Meerschaert, MM, A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model, Comput. Math. Appl., 66, 5, 693-701 (2013) · Zbl 1350.65092
[37] Zheng, X.; Wang, H., Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal. (2020)
[38] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM Numer. Anal., 47, 1760-1781 (2009) · Zbl 1204.26013
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