Abstract
In this study, we explore a new robust compact difference method (CDM) on graded meshes for the time-fractional nonlinear Kuramoto–Sivashinsky (KS) equation. This equation exemplifies a fourth-order sub-diffusion equation marked by nonlinearity. Considering the weak singularity often exhibited by exact solutions of time-fractional partial differential equations (TFPDEs) near the initial time, we introduce the L2-1\(_{\sigma }\) scheme on graded meshes to discretize the Caputo derivatives. By employing a novel double reduction order approach, we obtain a triple-coupled nonlinear system of equations. To address the nonlinear term \(uu_{x}\), we use a fourth-order nonlinear CDM, while the second and fourth derivatives in space are treated using the fourth-order linear CDM. We prove the solvability through Browder theorem. Additionally, \(\alpha \)-robust stability and convergence are demonstrated by introducing a modified discrete Grönwall inequality. Finally, we present numerical examples to corroborate the findings of our theoretical analysis.
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References
Ablowitz MJ, Ladik JF (1976) Nonlinear differential-difference equations and Fourier analysis. J Math Phys 17(6):1011–1018
Akrivis GD, Dougalis VA, Karakashian OA (1991) On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numerische Mathematik 59:31–53
Alikhanov AA (2015) A new difference scheme for the time fractional diffusion equation. J Comput Phys 280:424–438
Bhatt HP, Chowdhury A (2021) A high-order implicit-explicit Runge-Kutta type scheme for the numerical solution of the Kuramoto-Sivashinsky equation. Int J Comput Math 98(6):1254–1273
Chen H, Stynes M (2019) Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J Sci Comput 79:624–647
Chen L, Lu S, Xu T (2021) Fourier spectral approximation for time fractional Burgers equation with nonsmooth solutions. Appl Numer Math 169:164–178
Ersoy Hepson O (2021) Numerical simulations of Kuramoto-Sivashinsky equation in reaction-diffusion via Galerkin method. Math Sci 15(2):199–206
Han XL, Guo TAO, Nikan O, Avazzadeh Z (2023) Robust implicit difference approach for the time-fractional Kuramoto-Sivashinsky equation with the non-smooth solution. Fractals 2340061
Huang C, Stynes M (2021) \(\alpha \)-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation. Numer Algor 87:1749–1766
Huang C, Stynes M (2022) A sharp \(\alpha \)-robust \(L^{\infty }(H^1)\) error bound for a time-fractional Allen-Cahn problem discretised by the Alikhanov \(L2-1_{\sigma }\) scheme and a standard FEM. J Sci Comput 91(2):43
Huang C, Stynes M, Chen H (2021) An \(\alpha \)-robust finite element method for a multi-term time-fractional diffusion problem. J Comput Appl Math 389:113334
Jafari H, Daftardar-Gejji V (2006) Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition. Appl Math Comput 180(2):488–497
Kuramoto Y, Tsuzuki T (1976) Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog Theor Phys 55(2):356–369
Liao HL, McLean W, Zhang J (2021) A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem. Commun Comput Phys 30(2):567–601
Podlubny I (1999) Fractional differential equations, volume 198 of mathematics in science and engineering, pp 7–35
Qiu W, Chen H, Zheng X (2019) An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations. Math Comput Simul 166:298–314
Sahoo S, Ray SS (2015) New approach to find exact solutions of time-fractional Kuramoto-Sivashinsky equation. Physica A Stat Mech Appl 434:240–245
Sivashinsky GI (1983) Instabilities, pattern formation, and turbulence in flames. Annu Rev Fluid Mech 15(1):179–199
Sugimoto N (1991) Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. J Fluid Mech 225(631–653):5116
Sun ZZ, Zhang Q, Gao GH (2023) Finite difference methods for nonlinear evolution equations, 8th edn. De Gruyter, Berlin
Taneco-Hernández MA, Morales-Delgado VF, Gómez-Aguilar JF (2019) Fractional Kuramoto-Sivashinsky equation with power law and stretched Mittag-Leffler kernel. Physica A Stat Mech Appl 527:121085
Tian Q, Yang X, Zhang H, Xu D (2023) An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties. Comput Appl Math 42(6):246
Veeresha P, Prakasha DG (2021) Solution for fractional Kuramoto-Sivashinsky equation using novel computational technique. Int J Appl Comput Math 7(2):33
Wang X, Zhang Q, Sun Z (2021) The pointwise error estimates of two energy-preserving fourth-order compact schemes for viscous Burgers’ equation. Adv Comput Math 47:1–42
Yang X, Zhang H (2022) The uniform \(l^1\) long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data. Appl Math Lett 124(107644):203–224
Yang X, Zhang Z (2024) On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations. Appl Math Lett 150:108972
Yang X, Zhang H, Tang J (2021) The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions. Comput Math Appl 82:1–12
Yang X, Wu L, Zhang H (2023) A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Appl Math Comput 457:128192
Zhang Y, Feng M (2023) A local projection stabilization virtual element method for the time-fractional Burgers equation with high Reynolds numbers. Appl Math Comput 436:127509
Zhang H, Yang X, Xu D (2020) An efficient spline collocation method for a nonlinear fourth-order reaction subdiffusion equation. J Sci Comput 85:7
Zhang Q, Sun C, Fang ZW, Sun HW (2022) Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation. Appl Math Comput 418:126824
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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), Hunan Provincial Natural Science Foundation of China (2022JJ50083).
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Wang, J., Jiang, X., Yang, X. et al. A new robust compact difference scheme on graded meshes for the time-fractional nonlinear Kuramoto–Sivashinsky equation. Comp. Appl. Math. 43, 381 (2024). https://doi.org/10.1007/s40314-024-02883-4
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DOI: https://doi.org/10.1007/s40314-024-02883-4
Keywords
- Time-fractional Kuramoto–Sivashinsky equation
- L2-1\(_{\sigma }\) scheme
- Graded meshes
- Compact difference scheme
- Convergence and stability