Efficient numerical schemes for variable-order mobile-immobile advection-dispersion equation

Leqiang Zou; Yanzi Zhang; Leqiang Zou; Yanzi Zhang

doi:10.3934/nhm.2025018

Networks and Heterogeneous Media

2025, Volume 20, Issue 2: 387-405. doi: 10.3934/nhm.2025018

Previous Article Next Article

Research article Special Issues

Efficient numerical schemes for variable-order mobile-immobile advection-dispersion equation

Leqiang Zou ^{1
,
,},
Yanzi Zhang ²

1.
Henan College of Industry and Information Technology, Jiaozuo, Henan 454000, China
2.
Henan Jiaozuo Normal College, Jiaozuo, Henan 454000, China

Received: 19 January 2025 Revised: 14 April 2025 Accepted: 21 April 2025 Published: 25 April 2025

In this work, we present a high-order discontinuous Galerkin (DG) method with generalized alternating numerical fluxes to solve the variable-order (VO) fractional mobile-immobile advection-dispersion equation. This equation models complex transport phenomena where the order of differentiation varies with time, providing a more accurate representation of anomalous diffusion in heterogeneous media. For spatial and temporal discretization, the method employs the DG scheme and a finite difference method, respectively. Rigorous analysis confirms that the numerical scheme is unconditionally stable and convergent. Finally, numerical experiments are conducted to validate the theoretical results and illustrate the accuracy and efficiency of the scheme.
- fractional advection-dispersion equation,
- time fractional derivative,
- complex transport phenomena,
- convergent
Citation: Leqiang Zou, Yanzi Zhang. Efficient numerical schemes for variable-order mobile-immobile advection-dispersion equation[J]. Networks and Heterogeneous Media, 2025, 20(2): 387-405. doi: 10.3934/nhm.2025018

Related Papers:

Abstract

In this work, we present a high-order discontinuous Galerkin (DG) method with generalized alternating numerical fluxes to solve the variable-order (VO) fractional mobile-immobile advection-dispersion equation. This equation models complex transport phenomena where the order of differentiation varies with time, providing a more accurate representation of anomalous diffusion in heterogeneous media. For spatial and temporal discretization, the method employs the DG scheme and a finite difference method, respectively. Rigorous analysis confirms that the numerical scheme is unconditionally stable and convergent. Finally, numerical experiments are conducted to validate the theoretical results and illustrate the accuracy and efficiency of the scheme.

References

[1]	I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
[2]	W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1768–1777. https://doi.org/10.1016/j.na.2009.09.018 doi: 10.1016/j.na.2009.09.018
[3]	Q. Li, Y. Chen, Y. Huang, Y. Wang, Two-grid methods for nonlinear time fractional diffusion equations by L1-Galerkin FEM, Math. Comput. Simul., 185 (2021), 436–451. https://doi.org/10.1016/j.matcom.2020.12.033 doi: 10.1016/j.matcom.2020.12.033
[4]	X. Zheng, H. Wang, 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., 41 (2021), 1522–1545. https://doi.org/10.1093/imanum/draa013 doi: 10.1093/imanum/draa013
[5]	Y. Zhao, C. Shen, M. Qu, W. P. Bu, Y. F. Tang, Finite element methods for fractional diffusion equations, Int. J. Model., Simul., Sci. Comput., 11 (2020), 2030001. https://doi.org/10.1142/S1793962320300010 doi: 10.1142/S1793962320300010
[6]	X. Li, H. Rui, Two temporal second-order $H^{1}$-Galerkin mixed finite element schemes for distributed-order fractional sub-diffusion equations, Numerical Algorithms, 79 (2018), 1107–1130. https://doi.org/10.1007/s11075-018-0476-4 doi: 10.1007/s11075-018-0476-4
[7]	W. Bu, A. Xiao, W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput., 72 (2017), 422–441. https://doi.org/10.1007/s10915-017-0360-8 doi: 10.1007/s10915-017-0360-8
[8]	L. Feng, P. Zhuang, F. Liu, I. Turner, Y. Gu, Finite element method for space-time fractional diffusion equation, Numerical Algorithms, 72 (2016), 749–767. https://doi.org/10.1007/s11075-015-0065-8 doi: 10.1007/s11075-015-0065-8
[9]	A. J. Cheng, H. Wang, K. X. Wang, A Eulerian-Lagrangian control volume method for solute transport with anomalous diffusion, Numer. Methods Partial Differ. Equ., 31 (2015), 253–267. https://doi.org/10.1002/num.21901 doi: 10.1002/num.21901
[10]	M. Badr, A. Yazdani, H. Jafari, Stability of a finite volume element method for the time-fractional advection-diffusion equation, Numer. Methods Partial Differ. Equ., 34 (2018), 1459–1471. https://doi.org/10.1002/num.22243 doi: 10.1002/num.22243
[11]	F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Modell., 38 (2014), 3871–3878. https://doi.org/10.1016/j.apm.2013.10.007 doi: 10.1016/j.apm.2013.10.007
[12]	M. Stynes, E. O'Riordan, J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
[13]	R. Choudhary, S. Singh, P. Das, D. Kumar, A higher-order stable numerical approximation for time-fractional non-linear Kuramoto-Sivashinsky equation based on quintic B-spline, Math. Methods Appl. Sci., 47 (2024), 1–23. https://doi.org/10.1002/mma.9778 doi: 10.1002/mma.9778
[14]	X. M. Gu, H. W. Sun, Y. L. Zhao, X. C. Zheng, An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order, Appl. Math. Lett., 120 (2021), 107270. https://doi.org/10.1016/j.aml.2021.107270 doi: 10.1016/j.aml.2021.107270
[15]	X. D. Zhang, Y. L. Feng, Z. Y. Luo, J. Liu, A spatial sixth-order numerical scheme for solving fractional partial differential equation, Appl. Math. Lett., 159 (2025), 109265. https://doi.org/10.1016/j.aml.2024.109265 doi: 10.1016/j.aml.2024.109265
[16]	Y. Feng, X. Zhang, Y. Chen, L. Wei, A compact finite difference scheme for solving fractional Black-Scholes option pricing model, J. Inequal. Appl., 36 (2025), 36. https://doi.org/10.1186/s13660-025-03261-2 doi: 10.1186/s13660-025-03261-2
[17]	C. Li, F. Zeng, F. Liu, Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal., 15 (2012), 383–406. doi.org/10.2478/s13540-012-0028-x doi: 10.2478/s13540-012-0028-x
[18]	S. Guo, L. Mei, Z. Zhang, Y. Jiang, Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction-diffusion equation, Appl. Math. Lett., 85 (2018), 157–163. https://doi.org/10.1016/j.aml.2018.06.005 doi: 10.1016/j.aml.2018.06.005
[19]	X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108–2131. https://doi.org/10.1137/080718942 doi: 10.1137/080718942
[20]	A. Bhardwaj, A. Kumar, A meshless method for time fractional nonlinear mixed diffusion and diffusion-wave equation, Appl. Numer. Math., 160 (2021), 146–165. https://doi.org/10.1016/j.apnum.2020.09.019 doi: 10.1016/j.apnum.2020.09.019
[21]	Y. Gu, H. G. Sun, A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives, Appl. Math. Modell., 78 (2020), 539–549. https://doi.org/10.1016/j.apm.2019.09.055 doi: 10.1016/j.apm.2019.09.055
[22]	V. R. Hosseini, E. Shivanian, W. Chen, Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation, Eur. Phys. J. Plus, 130 (2015), 33. https://doi.org/10.1140/epjp/i2015-15033-5 doi: 10.1140/epjp/i2015-15033-5
[23]	Z. Avazzadeh, W. Chen, V. R. Hosseini, The coupling of RBF and FDM for solving higher order fractional partial differential equations, Appl. Mech. Mater., 598 (2014), 409–413. https://doi.org/10.4028/www.scientific.net/AMM.598.409 doi: 10.4028/www.scientific.net/AMM.598.409
[24]	P. Das, S. Rana, H. Ramos, A perturbation-based approach for solving fractional-order Volterra-Fredholm integro-differential equations and its convergence analysis, Int. J. Comput. Math., 97 (2020), 1994–2014. https://doi.org/10.1080/00207160.2019.1673892 doi: 10.1080/00207160.2019.1673892
[25]	P. Das, S. Rana, H. Ramos, Homotopy perturbation method for solving Caputo-type fractional-order Volterra-Fredholm integro-differential equations, Comput. Math. Methods, 1 (2019), e1047. https://doi.org/10.1002/cmm4.1047 doi: 10.1002/cmm4.1047
[26]	P. Das, S. Rana, H. Ramos, On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis, J. Comput. Appl. Math., 404 (2022), 113116. https://doi.org/10.1016/j.cam.2020.113116 doi: 10.1016/j.cam.2020.113116
[27]	L. Wei, Y. F. Yang, Optimal order finite difference/local discontinuous Galerkin method for variable-order time-fractional diffusion equation, J. Comput. Appl. Math., 383 (2021), 113129. https://doi.org/10.1016/j.cam.2020.113129 doi: 10.1016/j.cam.2020.113129
[28]	L. Wei, W. Li, Local discontinuous Galerkin approximations to variable-order time-fractional diffusion model based on the Caputo-Fabrizio fractional derivative, Math. Comput. Simul., 188 (2021), 280–290. https://doi.org/10.1016/j.matcom.2021.04.001 doi: 10.1016/j.matcom.2021.04.001
[29]	W. Li, L. Wei, Analysis of Local Discontinuous Galerkin Method for the Variable-order Subdiffusion Equation with the Caputo-Hadamard Derivative, Taiwanese J. Math., 28 (2024), 1095–1110. https://doi.org/10.11650/tjm/240801 doi: 10.11650/tjm/240801
[30]	Y. Liu, M. Zhang, H. Li, J. Li, High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional sub-diffusion equation, Comput. Math. Appl., 73 (2017), 1298–1314. https://doi.org/10.1016/j.camwa.2016.08.015 doi: 10.1016/j.camwa.2016.08.015
[31]	B. P. Moghaddam, J. A. T. Machado, Extended algorithms for approximating variable order fractional derivatives with applications, J. Sci. Comput., 71 (2016), 1351–1374. https://doi.org/10.1007/s10915-016-0343-1 doi: 10.1007/s10915-016-0343-1
[32]	L. Ramirez, C. Coimbra, On the selection and meaning of variable order operators for dynamic modeling, Int. J. Differ. Equ., 2010 (2010), 1–16. https://doi.org/10.1155/2010/846107 doi: 10.1155/2010/846107
[33]	Z. Chen, J. Z. Qian, H. B. Zhan, L. W. Chen, S. H. Luo, Mobile-immobile model of solute transport through porous and fractured media, IAHS Publ., 341 (2011), 154–158.
[34]	R. Schumer, D. A. Benson, M. M. Meerschaert, B. Baeumer, Fractal mobile/immobile solute transport, Water Resour. Res., 39 (2003), 1–12. https://doi.org/10.1029/2003WR002141 doi: 10.1029/2003WR002141
[35]	Y. Zhang, D. A. Benson, D. M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water Resour., 32 (2009), 561–581.
[36]	K. Sadri, H. Aminikhah, An efficient numerical method for solving a class of variable-order fractional mobile-immobile advection-dispersion equations and its convergence analysis, Chaos, Solitons Fractals, 146 (2021), 110896. https://doi.org/10.1016/j.chaos.2021.110896 doi: 10.1016/j.chaos.2021.110896
[37]	H. Ma, Y. Yang, Jacobi spectral collocation method for the time variable-order fractional mobile-immobile advection-dispersion solute transport model, East Asian J. Appl. Math., 6 (2016), 337–352. https://doi.org/10.4208/eajam.141115.060616a doi: 10.4208/eajam.141115.060616a
[38]	Z. G. Liu, A. J. Cheng, X. L. Li, A second order finite dfference scheme for quasilinear time fractional parabolic equation based on new fractional derivative, Int. J. Comput. Math., 95 (2017), 396–411. https://doi.org/10.1080/00207160.2017.1290434 doi: 10.1080/00207160.2017.1290434
[39]	A. Golbabai, O. Nikan, T. Nikazad, Numerical investigation of the time fractional mobile-immobile advection-dispersion model arising from solute transport in porous media, Int. J. Appl. Comput. Math., 5 (2019), 1–22. https://doi.org/10.1007/s40819-019-0635-x doi: 10.1007/s40819-019-0635-x
[40]	H. Zhang, F. Liu, M. S. Phanikumar, M. M. Meerschaert, A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model, Comput. Math. Appl., 66 (2013), 693–701. https://doi.org/10.1016/j.camwa.2013.01.031 doi: 10.1016/j.camwa.2013.01.031
[41]	W. Jiang, N. Liu, A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model, Appl. Numer. Math., 119 (2017), 18–32. https://doi.org/10.1016/j.apnum.2017.03.014 doi: 10.1016/j.apnum.2017.03.014
[42]	M. Saffarian, A. Mohebbi, An efficient numerical method for the solution of 2D variable order time fractional mobile-immobile advection-dispersion model, Math. Methods Appl. Sci., 44 (2021), 5908–5929. https://doi.org/10.1002/mma.7158 doi: 10.1002/mma.7158
[43]	C. F. M. Coimbra, Mechanica with variable-order differential operators, Ann. Phys., 12 (2003), 692–703. https://doi.org/10.1002/andp.200351511-1203 doi: 10.1002/andp.200351511-1203
[44]	Y. Cheng, X. Meng, Q. Zhang, Application of generalized Gauss-Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations, Math. Comp., 86 (2017), 1233–1267. https://doi.org/10.1090/mcom/3141 doi: 10.1090/mcom/3141
[45]	L. Wei, H. Wang, Y. Chen, Local discontinuous Galerkin method for a hidden-memory variable order reaction-diffusion equation, J. Appl. Math. Comput., 69 (2023), 2857–2872. https://doi.org/10.1007/s12190-023-01865-9 doi: 10.1007/s12190-023-01865-9
[46]	W. Wang, E. Barkai, Fractional advection-diffusion-asymmetry equation, Phys. Rev. Lett., 125 (2020), 240606. https://doi.org/10.1103/PhysRevLett.125.240606 doi: 10.1103/PhysRevLett.125.240606

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)