A space-time second-order method based on modified two-grid algorithm with second-order backward difference formula for the extended Fisher–Kolmogorov equation

References

• G. Ahlers and D.S. Cannell, Vortex-front propagation in rotating Couette-Taylor flow, Phys. Rev. Lett. 50 (1983), pp. 1583–1586.
   Web of Science ®Google Scholar
• D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math. 30 (1978), pp. 33–67.
   Web of Science ®Google Scholar
• H. Chen, D. Xu, J. Cao, and J. Zhou, A formally second order BDF ADI difference scheme for the three-dimensional time-fractional heat equation, Int J. Comput. Math. 97 (2020), pp. 1100–1117.
   Web of Science ®Google Scholar
• P. Coullet, C. Elphick, and D. Repaux, Nature of spatial chaos, Phys. Rev. Lett. 58 (1987), pp. 431–434.
   PubMed Web of Science ®Google Scholar
• P.E. Crouch and R. Grossman, Numerical integration of ordinary differential equations on mainfolds, J. Nonlinear. Sci. 3 (1993), pp. 1–33.
   Web of Science ®Google Scholar
• P. Danumjaya, A.K. Pany, and A.K. Pani, Morley FEM for the fourth-order nonlinear reaction-diffusion problems, Comput. Math. Appl. 22 (2021), pp. 229–245.
   Google Scholar
• G.T. Dee and W.V. Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett. 60 (1988), pp. 2641–2644.
   PubMed Web of Science ®Google Scholar
• E. Emmrich, Error of the two-step BDF for the incompressible Navier-Stokes problems, Math. Model. Numer. Anal. 38 (2004), pp. 757–764.
   Web of Science ®Google Scholar
• H. Fu, H. Guo, J. Hou, and J. Zhao, A stabilized mixed finite element method for steady and unsteady reaction-diffusion equations, Comput. Meth. Appl. Mech. Eng. 304 (2016), pp. 102–117.
   Web of Science ®Google Scholar
• C.W. Gear and L.R. Petzold, ODE methods for solution of differential/algebaric, SIAM J. Numer. Anal. 21 (1984), pp. 716–728.
   Web of Science ®Google Scholar
• D. He, On the L∞-norm convergence of a three-level linearly implicit finite difference method for the extended Fisher-Kolmogorov equation in both 1D and 2D, Comput. Math. Appl. 71 (2016), pp. 2594–2607.
   Web of Science ®Google Scholar
• A.T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equations, IMA J. Numer. Anal. 20 (2000), pp. 633–667.
   Web of Science ®Google Scholar
• R.M. Hornreich, M. Luban, and S. Shtrikman, Critical behaviour at the onset of k-space instability at the λ line, Phys. Rev. Lett. 35 (1975), pp. 1678–1681.
   Web of Science ®Google Scholar
• N. Khiari and K. Omrani, Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions, Comput. Math. Appl. 62 (2011), pp. 4151–4160.
   Web of Science ®Google Scholar
• X. Li, J. Shen, and H. Rui, Energy stability and convergence of SAV block-centered finite difference method for gradient flows, Math. Comput. 88 (2019), pp. 2047–2068.
   Web of Science ®Google Scholar
• Q. Li, Y. Chen, and Y. Huang, Two-grid methods for nonlinear time fractional diffusion equations by L1-Galerkin FEM, Math. Comput. Simul. 185 (2021), pp. 436–451.
   Web of Science ®Google Scholar
• C. Li, Y. Liu, and H. Li, Fast TT-M fourth-order compact difference schemes for a two-dimensional space fractional Gray-Scott model, Comput. Math. Appl. 141 (2023), pp. 191–206.
   Web of Science ®Google Scholar
• H. Li and Z. Zhang, Analysis of adaptive BDF2 scheme for diffusion equations, Math. Comput. 90 (2021), pp. 1207–1226.
   Web of Science ®Google Scholar
• W. Liu, J. Huang, and X. Long, Coupled nonlinear advection-diffusion-reaction system for prevention of groundwater contamination by modified upwind finite volume element method, Comput. Math. Appl. 69 (2015), pp. 477–493.
   Web of Science ®Google Scholar
• W. Liu and J. Cui, A two-grid block-centered finite difference algorithm for nonlinear compressible Darcy-Forchheimer model in porous media, J. Sci. Comput. 74 (2018), pp. 1786–1815.
   Web of Science ®Google Scholar
• Y. Liu and F. Liu, Time two-mesh algorithm combined with finite element method for time fractional water wave model, Int. J. Heat Mass Transf. 120 (2018), pp. 1132–1145.
   Web of Science ®Google Scholar
• W. Liu and Z. Sun, A block-centered finite difference method for reduced fracture model in Karst aquifer system, Comput. Math. Appl. 74 (2017), pp. 1455–1470.
   Web of Science ®Google Scholar
• R.K. Mohanty and D. Kaur, High accuracy compact operator methods for two-dimensional fourth order nonlinear parabolic partial differential equations, Comput. Methods Appl. Math. 17 (2017), pp. 617–641.
   Web of Science ®Google Scholar
• K. Nisar and S. Alsallami, On the exact solutions of nonlinear extended Fisher-Kolmogorov equation by using the He's variational approach, AIMS Math. 7 (2022), pp. 13874–13886.
   Web of Science ®Google Scholar
• Y. Niu, Y. Liu, and H. Li, Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media, Math. Comput. Simul. 203 (2023), pp. 387–407.
   Web of Science ®Google Scholar
• O. Oruc, An efficient wavelet collocation method for nonlinear two-space dimensional Fisher-Kolmogorov-Petrovsky-Piscounov equation and two-space dimensional extended Fisher-Kolmogorov equation, Eng. Comput. 36 (2020), pp. 839–856.
   Web of Science ®Google Scholar
• S. Phongthanapanich and P. Dechaumphai, Finite volume element method for analysis of unsteady reaction-diffusion problems, Acta. Mech. Sin. 25 (2009), pp. 481–489.
   Web of Science ®Google Scholar
• H. Rui and W. Liu, A two-grid block-centered finite difference method for Darcy-Forchheimer flow in porous media, SIAM J. Numer. Anal. 53 (2015), pp. 1941–1962.
   Web of Science ®Google Scholar
• H. Rui and H. Pan, A block-centered finite difference method for the Darcy-Forchheimer model, SIAM J. Numer. Anal. 50 (2012), pp. 2612–2631.
   Web of Science ®Google Scholar
• H. Rui and H. Pan, Block-centered finite difference methods for parabolic equation with time-dependent coefficient, Jpn. J. Ind. Appl. Math. 30 (2013), pp. 681–699.
   Web of Science ®Google Scholar
• W.V. Saarloos, Dynamical velocity selection: Marginal stability, Phys. Rev. Lett. 58 (1987), pp. 2571–2574.
   PubMed Web of Science ®Google Scholar
• W.V. Saarloos, Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection, Phys. Rev. Lett. A37 (1988), pp. 211–229.
   Google Scholar
• J. Xu, A new class of iterative methods for nonselfadjoint or indefinite elliptic problems, SIAM J. Numer. Anal. 29 (1992), pp. 303–319.
   Web of Science ®Google Scholar
• J. Xu, A novel two-grid method for semilinear equations, SIAM J. Sci. Comput. 15 (1994), pp. 231–237.
   Web of Science ®Google Scholar
• J. Xu, Two-grid discretization techniques for linear and non-linear PDEs, SIAM J. Numer. Anal. 33 (1996), pp. 1759–1777.
   Web of Science ®Google Scholar
• J. Xu, S. Xie, and H. Fu, A two-grid block-centered finite difference method for the nonlinear regularized long wave equation, Appl. Numer. Math. 171 (2022), pp. 128–148.
   Web of Science ®Google Scholar
• G. Zhu, Experiments on director waves in nematic liquid crystals, Phys. Rev. Lett. 49 (1982), pp. 1332–1335.
   Web of Science ®Google Scholar