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A weighted ADI scheme with variable time steps for diffusion-wave equations

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Abstract

We study a weighted alternating direction implicit (ADI) numerical method with variable time steps for two-dimensional diffusion-wave equations. The variable-step Alikhanov formula is employed to approximate the fractional derivatives in an equivalent coupled equations which is generated by the symmetric fractional-order reduction (SFOR) method. By adding a weighted small external term, we obtain a weighted ADI scheme for the diffusion-wave equations. The unconditional stability and convergence are analyzed by energy method, and the optimal temporal convergence order is \(\min \{2,\frac{3}{2}\alpha \}\), where \(1<\alpha <2\). The spatial compact scheme combined with the ADI method is also discussed. Numerical examples are provided to confirm the accuracy and efficiency of proposed schemes.

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References

  1. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, H., Stynes, M.: Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J. Sci. Comput. 79, 624–647 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, X., Di, Y., Duan, J., Li, D.: Linearized compact ADI schemes for nonlinear time-fractional Schrödinger equations. Appl. Math. Lett. 84, 160–167 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, X., Qin, H., Zhang, J.: A compact ADI scheme for two-dimensional fractional sub-diffusion equation with Neumann boundary condition. Appl. Numer. Math. 156, 50–62 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cui, M.: Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. J. Comput. Phys. 231, 2621–2633 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Du, R.L., Sun, Z.Z.: A fast temporal second-order compact ADI scheme for time fractional mixed diffusion-wave equations. East Asian J. Appl. Math. 11, 647–673 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fairweather, G., Yang, X., Da, Xu., Zhang, H.: An ADI Crank–Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation. J. Sci. Comput. 65, 1217–1239 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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, 650–678 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. 88, 2135–2155 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liao, H.L., Li, D., Zhang, J.: Sharp error estimate of a nonuniform L1 formula for time-fractional reaction–subdiffusion equations. SIAM J. Numer. Anal. 56, 1112–1133 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liao, H. L., Liu, N., Lyu, P.: Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models. SIAM J. Numer. Anal., 61, 2157–2181 (2023)

  12. Liao, H.L., McLean, W., Zhang, J.: A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57, 218–237 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liao, H.L., McLean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction–subdiffusion problem. Commun. Comput. Phys. 30, 567–601 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liao, H.L., Sun, Z.Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Part Differ. Equ. 26, 37–60 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liao, H.L., Tang, T., Zhou, T.: A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen–Cahn equations. J. Comput. Phys. 141, 109473 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liao, H.L., Yan, Y., Zhang, J.: Unconditional convergence of a two-level linearized fast algorithm for semilinear subdiffusion equations. J. Sci. Comput. 80, 1–25 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liao, H.L., Zhao, Y., Teng, X.: A weighted ADI scheme for subdiffusion equations. J. Sci. Comput. 69, 1144–1164 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lyu, P., Liang, Y., Wang, Z.: A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation. Appl. Numer. Math. 151, 448–471 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lyu, P., Vong, S.: A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations. J. Sci. Comput. 93, 34 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lyu, P., Vong, S.: Second-order and nonuniform time-stepping schemes for time fractional evolution equations with time-space dependent coefficients. J. Sci. Comput. 89, 49 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  22. McLean, K., Mustapha, K.: A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481–510 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Oldham, K., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  24. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  25. Qiao, L., Xu, D.: A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation. Adv. Comput. Math. 47, 64 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Saffarian, M., Mohebbi, A.: A novel ADI Galerkin spectral element method for the solution of two-dimensional time fractional subdiffusion equation. Int. J. Comput. Math. 98, 845–867 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sun, H., Sun, Z.Z.: A fast temporal second-order compact ADI difference scheme for the 2D multi-term fractional wave equation. Numer. Algorithms 86, 761–797 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, Y., Chen, H., Sun, T.: \(\alpha \)-Robust \(H^1\)-norm convergence analysis of ADI scheme for two-dimensional time-fractional diffusion equation. Appl. Numer. Math. 168, 75–83 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, Z., Cen, D., Mo, Y.: Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels. Appl. Numer. Math. 159, 190–203 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, Z., Liang, Y., Mo, Y.: A novel high order compact ADI scheme for two dimensional fractional integro-differential equations. Appl. Numer. Math. 167, 257–272 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, X., Wu, L., Zhang, H.: A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Appl. Math. Comput. 457, 128192 (2023)

    MathSciNet  MATH  Google Scholar 

  34. Zhai, S., Weng, Z., Feng, X., Yuan, J.: Investigations on several high-order ADI methods for time-space fractional diffusion equation. Numer. Algorithms 82, 69–106 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, J., Huang, J., Aleroev, T.S., Tang, Y.: A linearized ADI scheme for two-dimensional time-space fractional nonlinear vibration equations. Int. J. Comput. Math. 98, 2378–2392 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, W., Li, J., Yang, Y.: A fractional diffusion-wave equation with non-local regularization for image denoising. Signal Process. 103, 6–15 (2014)

    Article  Google Scholar 

  37. Zhang, Y., Sun, Z.: Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 230, 8713–8728 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang, Y., Sun, Z.: Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation. J. Sci. Comput. 59, 104–128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the two referees for their comments which improve the paper significantly.

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Authors and Affiliations

  1. School of Mathematics, Southwestern University of Finance and Economics, Chengdu, China

    Pin Lyu

  2. Department of Mathematics, University of Macau, Macao, China

    Seakweng Vong

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  1. Pin Lyu
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Correspondence to Seakweng Vong.

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Pin Lyu is supported by the NSF of China (12101510), the NSF of Sichuan Province (2022NSFSC1789) and Guanghua Talent Project of SWUFE.

Seakweng Vong is funded by the Science and Technology Development Fund, Macau SAR (File no. 0151/2022/A) and University of Macau (File no. MYRG2020-00035-FST, MYRG2022-00076-FST).

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Lyu, P., Vong, S. A weighted ADI scheme with variable time steps for diffusion-wave equations. Calcolo 60, 49 (2023). https://doi.org/10.1007/s10092-023-00543-3

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