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An efficient conservative splitting characteristic difference method for solving 2-d space-fractional advection–diffusion equations

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Abstract

In this paper, we develop an efficient splitting characteristic difference method for solving 2-dimensional two-sided space-fractional advection–diffusion equation. The intermediate numerical solutions are first computed by the piecewise parabolic method (PPM) where \(\bar{x}_i\) is solved by the explicit second-order Runge–Kutta scheme. Then, the interior solutions are computed by the splitting \(\sigma \)-implicit characteristic difference method. By some auxiliary lemmas, our scheme is proved stable in \(L^2\)-norm. The error estimate is given and we prove our schemes are of second-order convergence in space. Numerical experiments are used to verify our theoretical analysis.

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References

  • Berkowitz B, Scher H, Silliman S (2000) Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour Res 36:149–158

    Google Scholar 

  • Bermejo R (1995) A Galerkin-characteristic algorithm for transport-diffusion equations. SIAM J Numer Anal 32:425–454

    MathSciNet  MATH  Google Scholar 

  • Bromly M, Hinz C (2004) Non-Fickian transport in homogeneous unsaturated repacked sand. Water Resour Res 40:W07402

    Google Scholar 

  • Cao J, Xiao A, Bu W (2020) Finite difference/finite element method for tempered time fractional advection-dispersion equation with fast evaluation of Caputo derivative. J Sci Comput 48:1–29

    MathSciNet  MATH  Google Scholar 

  • Celik C, Duman M (2012) Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J Comput Phys 231:1743–1750

    MathSciNet  MATH  Google Scholar 

  • Chen M, Deng W (2014) Fourth order accurate scheme for the space fractional diffusion equations. SIAM J Numer Anal 52:1418–1438

    MathSciNet  MATH  Google Scholar 

  • Colella P, Woodward P (1984) The piecewise parabolic method (PPM) for gas-dynamical simulations. J Comput Phys 54:174–201

    MATH  Google Scholar 

  • Cushman J, Ginn T (2000) Fractional advection-dispersion equation: A classical mass balance with convolution-Fickian flux. Water Resour Res 36:3763–3766

    Google Scholar 

  • Dahle H, Ewing R, Russell T (1995) Eulerian-Lagrangian localized adjoint methods for a nonlinear advection-diffusion equation. Comput Methods Appl Mech Eng 122:223–250

    MathSciNet  MATH  Google Scholar 

  • Dawson C, Russell T, Wheeler M (1989) Some improved error estimates for the modified method of characteristics. SIAM J Numer Anal 26:1487–1512

    MathSciNet  MATH  Google Scholar 

  • Ding H, Li C (2017a) Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations. Fract Calc Appl Anal 20:722–764

    MathSciNet  MATH  Google Scholar 

  • Ding H, Li C (2017b) High-order numerical algorithms for Riesz derivatives via constructing new generating functions. J Sci Comput 71:759–784

    MathSciNet  MATH  Google Scholar 

  • Ding H, Li C (2017c) High-order algorithms for Riesz derivative and their applications (V). Numer Methods Partial Differ Equ 33:1754–1794

    MathSciNet  MATH  Google Scholar 

  • Ding H, Li C (2019) A high-order algorithm for time-caputo-tempered partial differential equation with Riesz derivatives in two spatial dimensions. J Sci Comput 80:81–109

    MathSciNet  MATH  Google Scholar 

  • Ding H, Li C (2020) Numerical algorithms for the time-Caputo and space-Riesz fractional Bloch-Torrey equations. Numer Methods PDE 36:772–799

    MathSciNet  Google Scholar 

  • Ding H, Li C, Chen Y (2015) High-order algorithms for Riesz derivative and their applications (II). J Comput Phys 293:218–237

    MathSciNet  MATH  Google Scholar 

  • Douglas J Jr, Russell T (1982) Numerical solution for convection-dominated diffusion problem based on combining the method of characteristics with finite element or differncen procedures. SIAM J Numer Anal 19:871–885

    MathSciNet  MATH  Google Scholar 

  • Fang Z, Zhang J, Sun H (2022) A fast finite volume method for spatial fractional diffusion equations on nonuniform meshes. Comput Math Appl 108:175–184

    MathSciNet  MATH  Google Scholar 

  • Feng L, Zhuang P, Liu F, Turner I, Gu Y (2016) Finite element method for space-time fractional diffusion equation. Numer Algorithm 72:749–767

    MathSciNet  MATH  Google Scholar 

  • Feng L, Zhuang P, Liu F, Turner I, Ahh V, Li J (2017) A fast second-order accurate method for a two-sided space-fractional diffusion equation with variable coefficients. Appl Math Comput 73:1155–1171

    MathSciNet  MATH  Google Scholar 

  • Fu K, Liang D (2016) The conservative characteristic FD methods for atmospheric aerosol transport problems. J Comput Phys 305:494–520

    MathSciNet  MATH  Google Scholar 

  • Fu K, Liang D (2017) The time second order mass conservative characteristic FDM for advection Cdiffusion equations in high dimensions. J Sci Comput 73:26–49

    MathSciNet  MATH  Google Scholar 

  • Fu K, Liang D (2019) A mass-conservative temporal second order and spatial fourth order characteristic finite volume method for atmosphertic pollution advection diffusion problems. SIAM J Sci Comput 41:1178–1210

    MathSciNet  Google Scholar 

  • Fu H, Liu H, Wang H (2019) A finite volume method for two-dimensional Riemann-Liouville space-fractional diffusion equation and its efficient implementation. J Comput Phys 388:316–334

    MathSciNet  MATH  Google Scholar 

  • Fu H, Chen Z, Liang X, Zhang B (2021) Efficient spatial second/fourth-order finite difference ADI methods for multi-dimensional variable-order time-fractional diffusion equations. Adv Comput Math 58:1–33

    MathSciNet  MATH  Google Scholar 

  • Gao G, Sun Z, Zhang H (2014) A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J Comput Phys 259:33–50

    MathSciNet  MATH  Google Scholar 

  • Han C, Wang Y, Li Z (2022) Novel patterns in a class of fractional reaction-diffusion models with the Riesz fractional derivative. Math Comput Simul 202:149–163

    MathSciNet  MATH  Google Scholar 

  • Hang T, Zhou Z, Pan H, Wang Y (2022) The conservative characteristic difference method and analysis for solving two-sided space-fractional advection-diffusion equations. Numer Algorithms. https://doi.org/10.1007/s11075-022-01363-2

    Article  Google Scholar 

  • Hejazi H, Moroney T, Liu F (2014) Stability and convergence of a finite volume method for the space fractional advection-dispersion equation. J Comput Appl Math 255:684–697

    MathSciNet  MATH  Google Scholar 

  • Jia J, Wang H (2015) A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh. J Comput Phys 299:842–862

    MathSciNet  MATH  Google Scholar 

  • Liang D, Wang W, Cheng Y (2009) An efficient second-order characteristic finite element method for nonlinear aerosol dynamic equations. Int J Numer Methods Eng 80:338–354

    MATH  Google Scholar 

  • Liu F, Zhuang P, Turner I, Burrage K, Anh V (2014) A new fractional finite volume method for solving the fractional diffusion equation. Appl Math Model 38:3871–3878

    MathSciNet  MATH  Google Scholar 

  • Liu Y, Du Y, Li H, Li J, He S (2015) A two-grid mixed finite element method for a nonlinear fourth order reaction-diffusion problem with time-fractional derivative. Comput Math Appl 70:2474–2492

    MathSciNet  MATH  Google Scholar 

  • Oldham K, Spanier J (1974) The fractional calculus. Academic Press, New York

    MATH  Google Scholar 

  • Ortigueira M (2006) Riesz potential operators and inverses via fractional centred derivatives. Int J Math Math Sci 2016:1–12

    MathSciNet  MATH  Google Scholar 

  • Ren J, Sun Z, Zhao X (2013) Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J Comput Phys 232:456–467

    MathSciNet  MATH  Google Scholar 

  • Simmons A, Yang Q, Moroney T (2017) A finite volume method for two-sided fractional diffusion equations on non-uniform meshes. J Comput Phys 335:747–759

    MathSciNet  MATH  Google Scholar 

  • Tadjeran C, Meerschaert M (2007) A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J Comput Phys 220:813–823

    MathSciNet  MATH  Google Scholar 

  • Tian W, Zhou H, Deng W (2015) A class of second order difference approximations for solving space fractional diffusion equations. Math Comput 84:1703–1727

    MathSciNet  MATH  Google Scholar 

  • Wang K, Wang H (2011) A fast characteristic finite difference method for fractional advection Cdiffusion equations. Adv Water Resour 34:810–816

    Google Scholar 

  • Wu Q, Huang J (2016) Fractional calculus. Tsinghua University Press, Beijing

    Google Scholar 

  • Yuste S, Quintana-Murillo J (2012) A finite difference scheme with non-uniform timesteps for fractional diffusion equations. Comput Phys Commun 183:2594–2600

    MathSciNet  MATH  Google Scholar 

  • Zhang X, Crawford J, Deeks LK et al (2005) A mass balance based numerical method for the fractional advection-dispersion equation: Theory and application. Water Resour Res 41:62–75

    Google Scholar 

  • Zhao Y, Bu W, Huang J, Liu D, Tang Y (2015) Finite element method for two-dimensional space-fractional advection-dispersion equations. Appl Math Comput 257:553–565

    MathSciNet  MATH  Google Scholar 

  • Zhao M, Wang H, Cheng A (2018) A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations with fractional derivative boundary conditions. J Sci Comput 74:1009–1033

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank two reviewers’ comments and suggestions on our manuscript which have helped to improve the paper greatly. This work was supported partially by Natural Science Foundation of Shandong Government (Grant Nos. ZR2021MA002, ZR2022MF239).

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Authors and Affiliations

  1. School of Information Science and Engineering, Shandong Agricultural University, Taian, 271018, Shandong, China

    Ning Wang, Xinxia Zhang, Zhongguo Zhou, Hao Pan & Yan Wang

Authors
  1. Ning Wang
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  2. Xinxia Zhang
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  3. Zhongguo Zhou
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  4. Hao Pan
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  5. Yan Wang
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Correspondence to Zhongguo Zhou.

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Wang, N., Zhang, X., Zhou, Z. et al. An efficient conservative splitting characteristic difference method for solving 2-d space-fractional advection–diffusion equations. Comp. Appl. Math. 42, 58 (2023). https://doi.org/10.1007/s40314-023-02198-w

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