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Journal of Computational Mathematics ›› 2020, Vol. 38 ›› Issue (4): 580-605.DOI: 10.4208/jcm.1902-m2017-0280

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HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY

Mohammad Tanzil Hasan, Chuanju Xu   

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen 361005, China
  • Received:2017-11-24 Revised:2018-12-05 Online:2020-07-15 Published:2020-07-15
  • Supported by:

    The research of this author is partially supported by NSF of China (51661135011 and 91630204).

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Mohammad Tanzil Hasan, Chuanju Xu. HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY[J]. Journal of Computational Mathematics, 2020, 38(4): 580-605.

In this paper, efficient numerical scheme is proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions are investigated.
Key words: Water waves, Nonlocal viscosity, Finite difference, Spectral method, Convergence order, Decay rate

CLC Number: 

[1] G.B. Whitman, Linear and Nonlinear Waves, J. Wiley, Newyork, 1974.

[2] T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive sysytems, Philos. Trans. R. Soc. London. Ser. A Math. Phys. Sci., 272:1220(1972), 47-78.

[3] L.A. Medeiros and G.P. Menzala, Existence and uniqueness for peridoc boundary solutions of the Benjamin-Bona-Mahony equation, SIAM Journal on Mathematical Analysis, 8:5(1977), 792-799.

[4] D.Dutykh and F. Dias, Viscous potential free-surface flows in a fluid layer of finite depth, Comptes Rendus Mathematique, 345:2(2007), 113-118.

[5] P.L.F. Liu and A. Orfila, Viscous effect on transient long-wave propagation, Fluid Mech., 520:1(2004), 83-92.

[6] J.L. Bona, M. Chen and J.C. Saut, Boussinesq equations and other systems for small amplitude long waves in nonlinear dispersive media I:Derivation and the linear theory, J. Nonlinear Sci. 12(2002), 283-318.

[7] T. Kakutani and K. Matsuuchi, Effect of viscosity on long gravity waves, J. Phys. Soc. Jpn., 39(1975), 237-246.

[8] M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with a nonlocal viscous dispersive term, Discrete cont. Dyn. Syst. -Ser. A, 27:4(2010), 1473-1492.

[9] E. Ott and R.N. Sudan, Damping of solitary waves, Phys. Fluids, 13(1970), 1432-1434.

[10] D. Dutkyh, Visco-potential free-surface flows and long wave modelling, Eur. J. Mech. - B/Fluids, Elsevier, 28:3(2009), 430-443.

[11] C.J. Amic, J.L. Bona and M.E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differ. Equ., 81:1(1989), 1-49.

[12] J.L. Bona, K.S. Promislow and C.E. Wayne, High order asymptotics of decaying solutions of some nonlinear, dispersive, dissipative wave equations, Nonlinearity, 8:6(1995), 1179.

[13] A. Dogan, Numerical solution of RLW equation using linear finite elements within Galerkins method, Appl. Math. Model., 26(2002), 771-783.

[14] K. Al-Khaled, S. Momani and A. Alawneh, Approximate wave solutions for generalized Benjamin -Bona-Mahony-Burgers equations, Appl. Math. Comput., 171(2005), 281-292.

[15] I. Dag, B. Saka and D. Irk, Galerkin method for the numerical solution of the RLW equation using quintic quadratic B-splines, J. Comput. Appl. Math., 190:1(2006), 532-547.

[16] K. Omrani, Convergence of fully discrete Galerkin approximations Benjamin -Bona-Mahony (BBM) equation, Appl. Math. Comput., 180(2006), 614-621.

[17] T. Achouri, N. Khiari and K. Omrani, On the convergence of diiference schemes for the Benjamin -Bona-Mahony (BBM) equation, Appl. Math. Comput., 182(2006), 999-1005.

[18] K. Omrani and M.Ayadi, Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation, Nonlinear Methods for Partial Differential Equations, 24:1(2008), 239-248.

[19] M.T. Hasan and C. Xu, The stability and convergence of time-stepping/spectral methods with asymptotic behaviour for the Rosenau-Burgers equation, Appl. Anal., doi:10.1080/00036811.2018. 1553034.

[20] O. Goubet and G. Warnault, Decay of solutions to a linear viscous asymptotic model for water waves, Chin. Ann. Math. -Ser B, 31:6(2010), 841-854.

[21] S. Dumont and J.B. Duval, Numerical investigation of the decay rate of solutions to models for water waves with nonlocal viscosity, Int. J. Number Anal. Model., 10:2(2013), 333-349.

[22] J. Zhang and C. Xu, Finite difference/spectral approximations to a water wave model with a nonlocal viscous term, Appl. Math. Model., 38(2014), 4912-4925.

[23] G.I. Jennings, Efficient numerical methods for water wave propagation in unbounded domains (Ph.D. thesis), Applied and Interdisciplinary Mathematics, The University of Michigan, 2012.

[24] I. Manoubi, Theoritical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative, Discrete cont. Dyn. Syst. -Ser. B, 19:9(2014), 2837-2863.

[25] Z. Mao and J. Shen, A semi-implicit spectral deferred correction method for two water wave models with nonlocal viscous term and numerical study of their decay rates, Sci. China Math., 8(2015), 1153-1168.

[26] C. Li and S. Zhaob, Efficient numerical schemes for fractional water wave models, Comput. Math. Appl., 71(2016), 238-254.

[27] J. Cao, C. Xu and Z. Wang, A high order finite difference/spectral approximations to the time fractional diffusion equations, Advanced Materials Research, 875-877(2014), 781-785.

[28] I. Podlubny, Fractional Differential Equations, Academic press, Sun Diego, 1999.

[29] C. Lv and C. Xu, Error analysis of a high order method for time fractional diffusion equations, SIAM J. Sci. Comput., 38:5(2016), A2699-A2724.

[30] A. Quarteroni and A. Valli, Numerical Approximation of the Partial Differential Equations, Springer-Verlag, Berlin, 1994.
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