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The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation

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Abstract

In the paper, we present a time two-grid difference (TGD) method for the approximation of two-dimensional (2D) nonlocal nonlinear wave equation (NNWE). First, the solution is obtained by solving a nonlinear system on the coarse grid (CG), then by using the numerical solutions obtained on the CG we construct a linearized system on the fine grid (FG). Meanwhile the auxiliary values calculated by Lagrange linear interpolation formula. Further, we prove the existence and uniqueness of solution on the CG and FG. Also the stability and convergence is proved strictly through the energy analysis scheme. Finally, two numerical examples are shown, which verify the proposed TGD method is more efficient than the general finite difference (GFD) method.

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References

  1. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2022)

    Google Scholar 

  2. Oldham, K., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier, Amsterdam (1974)

    Google Scholar 

  3. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, Amsterdam (1998)

    Google Scholar 

  4. Barenblatt, G.I., Zheltov, Y.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24(5), 852–864 (1960)

    Google Scholar 

  5. Ting, T.W.: A cooling process according to two-temperature theory of heat conduction. J. Math. Anal. Appl. 45(1), 23–31 (1974)

    MathSciNet  Google Scholar 

  6. Lei, Q., Pun, C.S.: Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems. J. Differ. Equ. 358, 339–385 (2023)

    MathSciNet  Google Scholar 

  7. Wang, R., Can, N.H., Nguyen, A.T., Tuan, N.H.: Local and global existence of solutions to a time-fractional wave equation with an exponential growth. Commun. Nonlinear Sci. Numer. Simul. 118, 107050 (2023)

    MathSciNet  Google Scholar 

  8. Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6), 23–28 (1996)

    MathSciNet  Google Scholar 

  9. Agrawal, O.P.: Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 29, 145–155 (2002)

    MathSciNet  Google Scholar 

  10. Chen, J., Liu, F., Anh, V., Shen, S., Liu, Q., Liao, C.: The analytical solution and numerical solution of the fractional diffusion-wave equation with dam. Appl. Math. Comput. 219(4), 1737–1748 (2012)

    MathSciNet  Google Scholar 

  11. Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Comput. Math. Appl. 64(10), 3377–3388 (2012)

    MathSciNet  Google Scholar 

  12. Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56(1), 1–23 (2018)

    MathSciNet  Google Scholar 

  13. Huang, C., An, N., Yu, X.: Unconditional energy dissipation law and optimal error estimate of fast L1 schemes for a time-fractional Cahn–Hilliard problem. Commun. Nonlinear Sci. Numer. Simul. 124, 107300 (2023)

    MathSciNet  Google Scholar 

  14. Zhang, H., Zeng, F., Jiang, X., Karniadakis, G.E.: Convergence analysis of the time-step numerical methods for time-fractional nonlinear subdiffusion equations. Fract. Calculus Appl. Anal. 25(2), 453–487 (2022)

    MathSciNet  Google Scholar 

  15. 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  Google Scholar 

  16. Joujehi, A.S., Derakhshan, M.H., Marasi, H.R.: An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis. Commun. Nonlinear Sci. Numer. Simul. 114, 106620 (2022)

    MathSciNet  Google Scholar 

  17. Alikhanov, A.A., Asl, M.S., Huang, C., Khibiev, A.: A second-order difference scheme for the nonlinear time-fractional diffusion-wave equation with generalized memory kernel in the presence of time delay. J. Comput. Appl. Math. 438, 115515 (2024)

    MathSciNet  Google Scholar 

  18. Lyu, P., Vong, S.: A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations. J. Sci. Comput. 93(1), 34 (2022)

    MathSciNet  Google Scholar 

  19. Tan, Z., Zeng, Y.: Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations. Appl. Math. Comput. 466, 128457 (2024)

    MathSciNet  Google Scholar 

  20. Li, K., Tan, Z.: Two-grid algorithms based on FEM for nonlinear time-fractional wave equations with variable coefficient. Comput. Math. Appl. 143, 119–132 (2023)

    MathSciNet  Google Scholar 

  21. Li, K., Tan, Z.: A two-grid fully discrete Galerkin finite element approximation for fully nonlinear time-fractional wave equations. Nonlinear Dyn. 111(9), 8497–8521 (2023)

    Google Scholar 

  22. Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15(1), 231–237 (1994)

    MathSciNet  Google Scholar 

  23. Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33(5), 1759–1777 (1996)

    MathSciNet  Google Scholar 

  24. Gu, Q., Chen, Y., Huang, Y.: Super convergence analysis of a two-grid finite element method for nonlinear time-fractional diffusion equations. Comput. Appl. Math. 41(8), 361 (2022)

    Google Scholar 

  25. Chen, Y., Wang, Y., Huang, Y., Fu, L.: Two-grid methods of expanded mixed finite-element solutions for nonlinear parabolic problems. Appl. Numer. Math. 144, 204–222 (2019)

    MathSciNet  Google Scholar 

  26. Li, X., Chen, Y., Chen, C.: An improved two-grid technique for the nonlinear time-fractional parabolic equation based on the block-centered finite difference method. J. Comput. Math. 40(3), 453 (2022)

    MathSciNet  Google Scholar 

  27. Chen, H., Qiu, W., Zaky, M.A., Hendy, A.S.: A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel. Calcolo 60(1), 13 (2023)

    MathSciNet  Google Scholar 

  28. Chen, H., Nikan, O., Qiu, W., Avazzadeh, Z.: Two-grid finite difference method for 1D fourth-order Sobolev-type equation with Burgers’ type nonlinearity. Math. Comput. Simul. 209, 248–266 (2023)

    MathSciNet  Google Scholar 

  29. Cen, D., Wang, Z.: Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations. Appl. Math. Lett. 129, 107919 (2022)

    MathSciNet  Google Scholar 

  30. Hu, H.: \(L_p\) error estimate of nonlinear Schrödinger equation using a two-grid finite element method. Numer. Methods Partial Differ. Equ. 39(4), 2865–2884 (2023)

    Google Scholar 

  31. Chen, C., Lou, Y., Hu, H.: Two-grid finite volume element method for the time-dependent Schrödinger equation. Comput. Math. Appl. 108, 185–195 (2022)

    MathSciNet  Google Scholar 

  32. Fu, H., Zhang, B., Zheng, X.: A high-order two-grid difference method for nonlinear time-fractional biharmonic problems and its unconditional \(\alpha \)-robust error estimates. J. Sci. Comput. 96(2), 54 (2023)

    MathSciNet  Google Scholar 

  33. Zhou, J., Yao, X., Wang, W.: Two-grid finite element methods for nonlinear time-fractional parabolic equations. Numer. Algorithms, 1–22 (2022)

  34. Li, K., Tan, Z.: Two-grid algorithms based on FEM for nonlinear time-fractional wave equations with variable coefficient. Comput. Math. Appl. 143, 119–132 (2023)

    MathSciNet  Google Scholar 

  35. Li, K., Tan, Z.: Two-grid fully discrete finite element algorithms on temporal graded meshes for nonlinear multi-term time-fractional diffusion equations with variable coefficient. Commun. Nonlinear Sci. Numer. Simul. 107360 (2023)

  36. Li, K., Tan, Z.: A two-grid fully discrete Galerkin finite element approximation for fully nonlinear time-fractional wave equations. Nonlinear Dyn. 111(9), 8497–8521 (2023)

    Google Scholar 

  37. Tan, Z., Li, K., Chen, Y.: A fully discrete two-grid finite element method for nonlinear hyperbolic integro-differential equation. Appl. Math. Comput. 413, 126596 (2022)

    MathSciNet  Google Scholar 

  38. Yang, J., Zhou, J., Nie, C.: A two-grid method for discontinuous Galerkin approximations to compressible miscible displacement problems. Comput. Math. Appl. 115, 57–67 (2022)

    MathSciNet  Google Scholar 

  39. Dai, X., Kuang, X., Xin, J., Zhou, A.: Two-grid based adaptive proper orthogonal decomposition method for time dependent partial differential equations. J. Sci. Comput. 84, 1–27 (2020)

    MathSciNet  Google Scholar 

  40. Zhang, J., Han, H., Yu, Y., Liu, J.: A new two-grid mixed finite element analysis of semi-linear reaction–diffusion equation. Comput. Math. Appl. 92, 172–179 (2021)

    MathSciNet  Google Scholar 

  41. Sun, Z.Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)

    MathSciNet  Google Scholar 

  42. Chen, H., Xu, D., Peng, Y.: An alternating direction implicit fractional trapezoidal rule type difference scheme for the two-dimensional fractional evolution equation. Int. J. Comput. Math. 92(10), 2178–2197 (2015)

    MathSciNet  Google Scholar 

  43. Sun, Z.Z.: Numerical Methods of Partial Differential Equations, 2nd edn. Science Press, Beijing (2012)

    Google Scholar 

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Authors and Affiliations

  1. School of Science, Hunan University of Technology, Zhuzhou, 412007, China

    Haixiang Zhang, Xiaoxuan Jiang, Furong Wang & Xuehua Yang

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  1. Haixiang Zhang
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  2. Xiaoxuan Jiang
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  3. Furong Wang
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  4. Xuehua Yang
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Correspondence to Xuehua Yang.

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The work was supported by National Natural Science Foundation of China Mathematics Tianyuan Foundation (12226337, 12226340, 12126321, 12126307), Scientific Research Fund of Hunan Provincial Education Department (21B0550, 22C0323), Hunan Provincial Natural Science Foundation of China (2022JJ50083, 2023JJ50164).

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Zhang, H., Jiang, X., Wang, F. et al. The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation. J. Appl. Math. Comput. 70, 1127–1151 (2024). https://doi.org/10.1007/s12190-024-02000-y

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  • DOI: https://doi.org/10.1007/s12190-024-02000-y

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