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A fast time two-mesh finite volume element algorithm for the nonlinear time-fractional coupled diffusion model

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Abstract

In this work, a fast second-order finite volume element (FVE) algorithm is proposed to solve the nonlinear time-fractional coupled diffusion model based on the time two-mesh (TT-M) computing method. In this algorithm, the integer and Riemann-Liouville fractional derivatives are approximated by the second-order backward difference formula and the WSGD formula respectively, the time interval is divided into coarse and fine meshes, then the three steps TT-M FVE algorithm is constructed by using the interpolation operator. The existence and uniqueness for the TT-M FVE algorithm are analyzed in detail, the asymptotically optimal a priori error estimates for variables u and v in the discrete \(L^{\infty }(L^{2}({\Omega }))\) and L2(H1(Ω) norms are obtained. It is shown that when time coarse and fine mesh sizes satisfy \(\tau _{c}=O(\tau _{f}^{1/2})\), the fast algorithm can achieve the same accuracy as the FVE algorithm, and reduce more computational cost. Finally, some numerical results are given to demonstrate the efficiency of the proposed algorithm.

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References

  1. Raposo, C.A., Sepúlveda, M., Villagrán, O. V., Pereira, D.C., Santos, M.L.: Solution and asymptotic behaviour for a nonlocal coupled system of reaction-diffusion. Acta Appl. Math. 102(1), 37–56 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Henry, B.I., Wearne, S.L.: Existence of Turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math. 62(3), 870–887 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Mommer, M.S., Lebiedz, D.: Modeling subdiffusion using reaction diffusion systems. SIAM J. Appl. Math. 70(1), 112–132 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Li, M., Huang, C.M., Zhao, Y.L.: Fast conservative numerical algorithm for the coupled fractional Klein-Gordon-Schrödinger equation. Numer. Algor. 84(3), 1081–1119 (2020)

    Article  MATH  Google Scholar 

  5. Zhang, G.Y., Huang, C.M., Li, M.: A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrödinger equations. Eur. Phys. J. Plus 133(4), 155 (2018)

    Article  Google Scholar 

  6. Eslami, M.: Exact traveling wave solutions to the fractional coupled nonlinear Schrödinger equations. Appl. Math. Comput. 285, 141–148 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Hussein, A.J.: A weak Galerkin finite element method for solving time-fractional coupled Burgers’ equations in two dimensions. Appl. Numer. Math. 156, 265–275 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guo, S.M., Mei, L.Q., Hou, Y.R., Zhang, Z.Q.: An efficient finite difference/Hermite-Galerkin spectral method for time-fractional coupled sine-Gordon equations on multidimensional unbounded domains and its application in numerical simulations of vector solitons. Comput. Phys. Commun. 237, 110–128 (2019)

    Article  MathSciNet  Google Scholar 

  9. Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, M.H., Deng, W.H.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52(3), 1418–1438 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    MathSciNet  MATH  Google Scholar 

  13. Liao, H.L., Li, D.F., Zhang, J.W.: Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 56, 1112–1133 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Feng, L.B., Liu, F.W., Turner, I.: Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains. Commun. Nonlinear Sci. Numer. Simul. 70, 354–371 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, M., Shi, D.Y., Pei, L.F.: Convergence and superconvergence analysis of finite element methods for the time fractional diffusion equation. Appl. Numer. Math. 151, 141–160 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, C.P., Li, D.X., Wang, Z.: L1/LDG method for the generalized time-fractional Burgers equation. Math. Comput. Simul. 187, 357–378 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jiang, Y.J., Xu, X.J.: A monotone finite volume method for time fractional Fokker-Planck equations. Sci. China Math. 62(4), 783–794 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hou, Y.X., Feng, R.H., Liu, Y., Li, H., Gao, W.: A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system. Int. J. Model. Simul. Sci. Comput. 8(1), 1750012 (2017)

    Article  Google Scholar 

  21. Kumar, D., Chaudhary, S., Kumar, V.V.K.S.: Finite element analysis for coupled time-fractional nonlinear diffusion system. Comput. Math. Appl. 78, 1919–1936 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo, S.M., Yan, W.J., Mei, L.Q., Wang, Y., Wang, L.L.: A linearized spectral-Galerkin method for three-dimensional Riesz-like space fractional nonlinear coupled reaction-diffusion equations. Numer. Math. Theor. Meth. Appl. 14(3), 738–772 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Heydari, M.H., Avazzadeh, Z., Atangana, A.: Orthonormal shifted discrete Legendre polynomials for solving a coupled system of nonlinear variable-order time fractional reaction-advection-diffusion equations. Appl. Numer. Math. 161, 425–436 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Feng, R.H., Liu, Y., Hou, Y.X., Li, H., Fang, Z.C.: Mixed element algorithm based on a second-order time approximation scheme for a two-dimensional nonlinear time fractional coupled sub-diffusion model. Eng. Comput. 38, 51–68 (2022)

    Article  Google Scholar 

  25. Hendy, A.S., Zaky, M.A.: Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations. Eng. Comput. 38, 1351–1363 (2022)

    Article  Google Scholar 

  26. Liu, Y., Yu, Z.D., Li, H., Liu, F.W., Wang, J.F.: Time two-mesh algorithm combined with finite element method for time fractional water wave model. Int. J. Heat Mass Transfer. 120, 1132–1145 (2018)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, Z.B., Vong, S.W.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1–15 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yin, B.L., 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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, Y., Fan, E.Y., Yin, B.L., Li, H., Wang, J.F.: TT-M finite element algorithm for a two-dimensional space fractional Gray-Scott model. Comput. Math. Appl. 80, 1793–1809 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Qiu, W.L., Xu, D., Guo, J., Zhou, J.: A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. Numer. Algor. 85, 39–58 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu, D., Guo, J., Qiu, W.L.: Time two-grid algorithm based on finite difference method for two-dimensional nonlinear fractional evolution equations. Appl. Numer. Math. 152, 169–184 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  33. Niu, Y.X., Liu, Y., Li, H., Liu, F.W.: Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media. Math. Comput. Simulat. https://doi.org/10.1016/j.matcom.2022.07.001

  34. Wen, C., Liu, Y., Yin, B.L., Li, H., Wang, J.F.: Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model. Numer. Algor. 88, 523–553 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, R.H., Chen, Z.Y., Wu, W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000)

    Book  MATH  Google Scholar 

  36. Ewing, R., Lazarov, R., Lin, Y.P.: Finite volume element aproximations of nonlocal reactive flows in porous media. Numer. Meth. Part. D. E. 16 (3), 285–311 (2000)

    Article  MATH  Google Scholar 

  37. Chou, S.H., Li, Q.: Error estimates in L2, H1 and \(L^{\infty }\) in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comp. 69, 103–120 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Chou, S.H., Kwak, D.Y., Li, Q.: Lp error estimates and superconvergence for covolume or finite volume element methods. Numer. Meth. Part. D. E. 19, 463–486 (2003)

    Article  MATH  Google Scholar 

  39. Sinha, R.K., Ewing, R.E., Lazarov, R.D.: Some new error estimates of a semidiscrete finite volume element method for a parabolic integro-differential equation with nonsmooth initial data. SIAM J. Numer. Anal. 43(6), 2320–2344 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Chen, C.J., Liu, W.: A two-grid finite volume element method for a nonlinear parabolic problem. Int. J. Numer. Anal. Mod. 12(2), 197–210 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Sayevand, K., Arjang, F.: Finite volume element method and its stability analysis for analyzing the behavior of sub-diffusion problems. Appl. Math. Comput. 290, 224–239 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Karaa, S., Pani, A.K.: Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data. ESAIM: M2AN 52, 773–801 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Karaa, S., Mustapha, K., Pani, A.K.: Finite volume element method for two-dimensional fractional subdiffusion problems. IMA J. Numer. Anal. 37(2), 945–964 (2017)

    MathSciNet  MATH  Google Scholar 

  44. Fang, Z.C., Zhao, J., Li, H., Liu, Y.: Finite volume element methods for two-dimensional time fractional reaction-diffusion equations on triangular grids. Appl. Anal. https://doi.org/10.1080/00036811.2022.2027374

  45. Browder, F.E.: Existence and uniqueness theorems for solutions of nonlinear boundary value problems. Proc. Sympos. Appl. Math. 17, 24–49 (1965)

    Article  MathSciNet  Google Scholar 

  46. Liu, Y., Du, Y.W., Li, H., Wang, J.F.: A two-grid finite element approximation for a nonlinear time-fractional Cable equation. Nonlinear Dyn. 85, 2535–2548 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Xu, J.C.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the editors and reviewers for their helpful comments and suggestions on improving the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (11701299,12161063), the Natural Science Foundation of Inner Mongolia (2020MS01003, 2021MS01018), the Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region (NMGIRT2207), and the Central Government Guided Local Science and Technology Development Fund Project of China.

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  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Zhichao Fang, Hong Li & Yang Liu

  2. School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot, 010070, China

    Jie Zhao

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Correspondence to Hong Li.

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Fang and Zhao wrote the main manuscript text, Li and Liu discussed the numerical theories in the manuscript. All authors worked together and contributed to this work; all authors reviewed and approved the final manuscript.

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Fang, Z., Zhao, J., Li, H. et al. A fast time two-mesh finite volume element algorithm for the nonlinear time-fractional coupled diffusion model. Numer Algor 93, 863–898 (2023). https://doi.org/10.1007/s11075-022-01444-2

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