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Original Articles

A high-order implicit–explicit Runge–Kutta type scheme for the numerical solution of the Kuramoto–Sivashinsky equation

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Pages 1254-1273 | Received 10 Jul 2020, Accepted 14 Aug 2020, Published online: 11 Sep 2020

References

  • G.D. Akrivis, Finite difference discretization of the Kuramoto–Sivashinsky equation. Numer. Math.  63 (1992), pp. 1–11.
  • G.D. Akrivis, Finite element discretization of the Kuramoto–Sivashinsky equation, Numer. Anal. Math. Model. 29(1) (1994), pp. 155–163.
  • G. Akrivis and Y.-S Smyrlis, Implicit-explicit BDF methods for the Kuramoto–Sivashinsky equation, Appl. Numer. Math. 51 (2004), pp. 159–161.
  • S.V. Alekseenko, V.E. Nakoryakov, and B.G. Pokusaev, Wave formation on vertical falling liquid films, Int. J. Multphase Flow. 11 (1985), pp. 607–627.
  • E.O. Asante-Asanami, A. Kleefeld, and B.A. Wade, A second-order exponential time differencing scheme for non-linear reaction–diffusion systems with dimensional splitting, J. Comput. Phys. 415 (2020), p. 109490.
  • D.J. Benny, Long waves in liquid film, J. Math. Phys. 45 (1966), pp. 150–155.
  • G. Beylkin, J.M. Keiser, and L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comput. Phys. 147 (1998), pp. 362–387.
  • H.P. Bhatt and A.Q.M. Khaliq, Higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equations, Appl. Math. Comput. 228 (2014), pp. 271–291.
  • H.P. Bhatt and A.Q.M. Khaliq, The locally extrapolated exponential time differencing LOD scheme for multidimensional reaction–diffusion systems, J. Comput. Appl. Math. 285 (2015), pp. 256–278.
  • M. Caliari and A. Ostermann, Implementation of exponential Rosenbrock type integrals, App. Numer. Math. 59 (2009), pp. 568–581.
  • C.I. Christov and K.L. Bekyarov, A Fourier-series method for solving solution problems, SIAM J. Sci. Stat. Comput. 11(4) (1990), pp. 631–647.
  • C.I. Christov and M.G. Velarde, Dissipative solitons, Phys. D. 86 (1995), pp. 323–347.
  • P. Clavin, Dynamic behavior of premixed flame fronts in laminar and turbulent flows, Prog. Energy Combust. Sci. 11(1) (1985), pp. 1–59.
  • S.M. Cox and P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002), pp. 430–455.
  • B. Fornberg and T.A. Driscoll, A fast spectral for nonlinear wave equations with linear dispersion, J. Comput. Phys. 155(2) (1999), pp. 456–467.
  • U. Frisch, Z.S. She, and O. Thual, Viscoelastic behaviour of cellular solutions to the Kuramoto–Sivashinsky model, J. Fluid. Mech. 168 (1986), pp. 221–240.
  • M. Hochburck and C. Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal. 34 (1997), pp. 1911–1925.
  • G.M. Homsy, Model equations for wavy viscous film flow, Lect. Appl. Math. 15 (1974), pp. 191–194.
  • J.M. Hyman and B. Nicolaenko, The Kuramoto–Sivashinsky equation: A bridge between PDE's and dynamical systems, Phys. D 18 (1986), pp. 113–126.
  • P.L. Kapitza, Wave flow in thin layer of viscous liquid, Zh. Exp. Theor. Fiz. 18 (1948), pp. 3–18.
  • A.K. Kassam and L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput.26(4) (2005), pp. 1214–1233.
  • A.H. Khater and R.S. Temsah, Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods, Comput. Math. Appl. 56 (2008), pp. 1465–1472.
  • D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 39 (1895), pp. 422–443.
  • S. Krogstad, Generalized integrating factor methods for stiff PDEs, J. Comput. Phys. 203 (2005), pp. 72–88.
  • Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys. 55 (1976), p. 356–369.
  • H. Lai and C.F. Ma, Lattice Boltzmann method for the generalized Kuramoto–Sivashinsky equation, Phys. A 388 (2009), pp. 1405–1412.
  • M. Lakestania and M. Dehghan, Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions, Appl. Math. Model. 36 (2012), pp. 605–617.
  • S.K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103(1) (1992), pp. 16–42.
  • A.V. Manickam, K.M. Moudgalya, and A.K. Pani, Second-order splitting combined with orthogonal cubic spline collocation method for the Kuramoto–Sivashinsky equation, Comput. Math. Appl. 35 (1998), pp. 5–25.
  • R.C. Mittal and G. Arora, Quintic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation, Comm. Nonlinear Sci. Numer. Simul. 15 (2010), pp. 2798–2808.
  • C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev. 45 (2003), pp. 3–49.
  • A. Mouloud, H. Fellouah, B.A. Wade, and M. Kessal, Time discretization and stability regions for dissipative-dispersive Kuramoto–Sivashinsky equation arising in turbulent gas flow over laminar liquid, J. Comput. Appl. Math. 330 (2018), pp. 605–617.
  • H. Otomo, B.M. Boghosian, and F. Dubois, Efficient lattice Boltzmann models for Kuramoto–Sivashinsky equation, Comput. Fluids. 172 (2018), pp. 683–688.
  • A. Pumir, P. Manneville, and Y. Pomaeu, On the solitary waves running down an inclined plane, J. Fluid Mech. 135 (1983), pp. 27–50.
  • B.K. Singh, G. Arora, and P. Kumar, A note on solving the fourth-order Kuramoto–Sivashinsky equation by the compact finite difference scheme, Ain Shams Eng. J. 9 (2018), pp. 1581–1589.
  • T. Shlang and G.I. Sivashinsky, Irregular flow of a liquid film down a vertical column, J. Phys. Paris43 (1982), pp. 459–466.
  • G.I. Sivashinsky, Nonlinear analysis of hydrodynamics instability in laminar flames – I. derivation of basic equations, Acta Astronaut. 4 (1977), pp. 1177–1206.
  • G.I. Sivashinsky, Instabilities, pattern-formation, and turbulence in flames, Ann. Rev. Fluid Mech. 15 (1983), pp. 179–199.
  • G.I. Sivashinsky and D.M. Michelson, On irregular wavy flow of a liquid film down a vertical plane, Prog. Theor. Phys. 63 (1980), pp. 2112–2114.
  • Y.A. Suhov, A spectral method for the time evolution in parabolic problems, J. Sci. Comput. 29 (2006), pp. 201–217.
  • H. Tal-Ezer, On restart and error estimation for Krylov approximation of w=f(A)v, SIAM J. Sci. Comput. 29 (2007), pp. 2426–2441.
  • M. Uddin, S. Haq, Siraj-ul-Islam, A mesh-free numerical method for solution of the family of Kuramoto–Sivashinsky equations, Appl. Math. Comput. 212(2) (2009), pp. 458–469.
  • Y. Xu and C.W. Shu, Local discontinuous Galerkin methods for the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations, Comput. Meth. Appl. Mech. Eng. 195 (2006), pp. 3430–3447.
  • M. Zarebnia and R. Parvaz, Septic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation, Int. J. Math. Comput. Sci. Eng. 2(1) (2013), pp. 55–61.

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