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An efficient computational approach for two-dimensional variant of nonlinear-dispersive model of shallow water wave


A three-level linearized difference scheme for two-dimensional dispersive shallow water wave that is governed by the Rosenau-RLW equation is considered. It is proved that the proposed difference scheme is conservative, uniquely solvable and unconditionally convergent. The convergence order in maximum norm is \(O(\tau ^2+h_1^2+h_2^2)\), where \(\tau\) is the temporal grid size and \(h_1, h_2\) are spatial grid sizes in the x- and y-directions, respectively. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and other existing method. Comparison reveals that our method improves the accuracy of the space and time direction and shortens computation time largely.

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  1. 1.

    Göktas U, Hereman W (1998) Computation of conservation laws for nonlinear lattices. Phys D 123(1–4):425–436

  2. 2.

    Hereman W (2006) Symbolic computation of conservation laws of nonlinear partial differential equations in multidimensions. Int J Quantum Chem 106(1):278–299

  3. 3.

    Kara AH (2009) A symmetry invariance analysis of the multipliers and conservation laws of the Jaulent–Miodek and families of systems of KdV-type equations. J Nonlinear Math Phys 16:149–156

  4. 4.

    Wang Y-Y, Dai CQ (2013) Elastic interaction between multivalued foldons and anti-foldons for the (2+1)-dimensional variable coefficient Brauer–Kaup system in water waves. Nonlinear Dyn 74(1–2):429–438

  5. 5.

    Zhong WP, Belic M (2013) Resonance solitons produced by azimuthal modulation in self-focusing and self-defocussing materials. Nonlinear Dyn 73(4):2091–2102

  6. 6.

    Razborova P, Triki H, Biswas A (2013) Perturbation of dispersive shallow water waves. Ocean Eng 63:1–7

  7. 7.

    Razborova P, Moraru L, Biswas A (2014) Perturbation of dispersive shallow water waves with Rosenau-KdV-RLW equation with power law nonlinearity. Rom J Phys 59:658–676

  8. 8.

    Rosenau P (1986) A quasi-continuous description of a non-linear transmission line. Phys Scr 34:827–829

  9. 9.

    Rosenau P (1988) Dynamics of dense discrete systems. Progr Theor Phys 79:1028–1042

  10. 10.

    Park MA (1990) On the Rosenau equation. Math Appl Comput 9:145–152

  11. 11.

    Atouani N (2013) Khaled Omrani Galerkin finite element method for the Rosenau-RLW equation. Comput Math Appl 66:289–303

  12. 12.

    Chung SK (2001) Numerical methods for the Rosenau equation. Appl Anal 77:351–369

  13. 13.

    Kim YD, Lee HY (1998) The convergence of finite element Galerkin solution of the Rosenau equation. Korean J Comput Appl Math 5:171–180

  14. 14.

    Omrani K, Abidi F, Achouri T, Khiari N (2008) A new conservative finite difference scheme for the Rosenau equation. Appl Math Comput 201:35–43

  15. 15.

    Atouani N, Omrani K (2015) A new conservative high-order accurate difference scheme for the Rosenau equation. Appl Anal 94:2435–2455

  16. 16.

    Peregrine DH (1966) Calculations of the development of an unduiar bore. J Fluid Mech 25:321–330

  17. 17.

    Razborova P, Kara AH, Biswas A (2015) Additional conservation laws for Rosenau-KdV-RLW equation with power law nonlinearity by Lie symmetry. Nonlinear Dyn 79:743–748

  18. 18.

    Triki H, Turgut AK, Moshokoa S et al (2016) Soliton solutions to KdV equation with spatio-temporal dispersion. Ocean Eng 114:192–203

  19. 19.

    Biswas A, Triki H, Labidi M (2011) Bright and dark solitons of the Rosenau-Kawahara equation with power law nonlinearity. Phys Wave Phenom 19(1):24–29

  20. 20.

    Wazwaz AM (2009) Multiple soliton solutions and multiple- singular soliton solutions for two higher-dimensional shallow water wave equations. Appl Math Comput 211:495–501

  21. 21.

    Wazwaz AM (2009) Multiple soliton solutions and multiple-singular soliton solutions for (2+1)-dimensional shallow water wave equations. Phys Lett A 37:2927–2930

  22. 22.

    Wazwaz AM (2010) Multiple-soliton solutions for extended shallow water wave equations. Stud Math Sci 1:21–29

  23. 23.

    Karakoc SBG, Gao F, Bhowmik SK (2018) Solitons and shock waves solutions for the Rosenau-KdV-RLW equation. J Sci Arts 4(45):1073–1088

  24. 24.

    Ak T, GaziKarako SB, Triki H (2016) Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation. Eur Phys J Plus Sayi 131:1–15

  25. 25.

    Karakoc SBG, Ak T (2016) Numerical simulation of dispersive shallow water waves with Rosenau-KdV equation. Int J Adv Appl Math Mech 3:32–40

  26. 26.

    Karakoc SBG (2018) A detailed numerical study on generalized Rosenau-KdV equation with finite element method. J Sci Arts 4(45):837–852

  27. 27.

    Omrani K, Ayadi M (2008) Finite difference discretization of the Benjamin-Bona-Mahony-Burgers (BBMB) equation. Numer Methods Partial Differ Equ 24(1):239–248

  28. 28.

    Rouatbi A, Omrani K (2017) Two conservative difference schemes for a model of nonlinear dispersive equations Chaos. Solitons Fractals 104:516–530

  29. 29.

    Rouatbi A, Achouri T, Omrani K (2018) High-order conservative difference scheme for a model of nonlinear dispersive equations. Comput Appl Math 37:4169–4195.

  30. 30.

    Ghiloufi A, Rouatbi A, Omrani K (2018) A new conservative fourth-order accurate difference scheme for solving a model of nonlinear dispersive equations. Math Methods Appl Sci. 41:5230–5253.

  31. 31.

    Ghiloufi A, Omrani K (2017) New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves. Numer Methods Partial Differ 34:451–500.

  32. 32.

    He D (2016) Exact solitary solution and a three-level linearly implicit conservative finite difference method for the generalized Rosenau-Kawahara-RLW equation with generalized Novikov type perturbation. Nonlinear Dyn 85(1):479–498

  33. 33.

    Pan X, Zhang L (2012) On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation. Appl Math Model 36:3371–3378

  34. 34.

    Ghiloufi A, Kadri T (2017) Analysis of new conservative difference scheme for two-dimensional Rosenau-RLW equation. Appl Anal 96(7):1255–1267

  35. 35.

    Zhou YL (1990) Applications of discrete functional analysis to the finite difference method. International Academic Publishers, Beijing

  36. 36.

    Piao G-R, Lee J-Y, Cai G-X (2016) Analysis and computational method based on quadratic B-spline FEM for the Rosenau-Burgers equation. Numer Methods Partial Differ Equ 32:877–895

  37. 37.

    Chung SK, Pani AK (2001) Numerical methods for the Rosenau equation. Appl Anal 77:351–369

  38. 38.

    Koley U (2012) Error estimates for a fully discrete spectral scheme for Korteweg-de Vries-Kawahara equation. Cent Eur J Math 10:173–187

  39. 39.

    Iório RJ Jr (1986) On the Cauchy problem for the Benjamin–Ono equation. Commun Partial Differ Equ 11:1031–1081

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We would like to thank the reviewers that their comments and suggestions have really improved the quality of the paper.

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Correspondence to Khaled Omrani.

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Lemma 1

For any grid functions \(V^n\in {\mathcal {V}}_h,\) we have

$$\begin{aligned} ({\partial }_{{\bar{t}}}V^n,{\bar{V}}^n)& = \frac{1}{2} \partial _{{\bar{t}}}\Vert V^n\Vert ^2, \end{aligned}$$
$$\begin{aligned} (\Delta _h(\partial _{{\bar{t}}}V^n), {\bar{V}}^n)& = -\frac{1}{2} \partial _{{\bar{t}}}|V^n|_1^2, \end{aligned}$$
$$\begin{aligned} (\Delta _h^2(\partial _{{\bar{t}}}V^n), {\bar{V}}^n)& = \frac{1}{2} \partial _{{\bar{t}}}\Vert \Delta _hV^n\Vert ^2, \end{aligned}$$
$$\begin{aligned} ({\nabla }_h{\bar{V}}^n,{\bar{V}}^n)& = 0, \end{aligned}$$
$$\begin{aligned} (\phi (V^n,\bar{V^n}),{\bar{V}}^n)& = 0. \end{aligned}$$


For every \(V^n\in {\mathcal {V}}_h,\) we have

$$\begin{aligned} ({\partial }_{{\bar{t}}}V^n,{\bar{V}}^n)& = h_1h_2\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}\displaystyle \frac{V_{i,j}^{n+1}- V_{i,j}^{n-1}}{2\tau }\cdot \displaystyle \frac{V_{i,j}^{n+1}+ V_{i,j}^{n-1}}{2}\\& = \displaystyle \frac{h_1h_2}{2}\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2} \displaystyle \frac{(V_{i,j}^{n+1})^2- (V_{i,j}^{n-1})^2}{2\tau }\\& = \displaystyle \frac{1}{2} \partial _{{\bar{t}}}\Vert V^n\Vert ^2, \end{aligned}$$

and (A.1) follows. In view of difference properties and (2.5), we obtain for \(V^n\in {\mathcal {V}}_h\)

$$\begin{aligned} (\Delta _h({\partial }_{{\bar{t}}}V^n),{\bar{V}}^n)& = \displaystyle \frac{h_1h_2}{4\tau }\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}(\delta ^2_x+\delta ^2_y)({V_{i,j}^{n+1}- V_{i,j}^{n-1}})\\&\cdot ({V_{i,j}^{n+1}+ V_{i,j}^{n-1}})\\& = -\displaystyle \frac{h_1h_2}{4\tau }\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2} (\delta _x+\delta _y)({V_{i,j}^{n+1}- V_{i,j}^{n-1}})\\&\cdot (\delta _x+\delta _y)({V_{i,j}^{n+1}+ V_{i,j}^{n-1}})\\& = -\displaystyle \frac{1}{4\tau }(|V^{n+1}|_1^2-|V^{n-1}|_1^2) \\& = -\displaystyle \frac{1}{2} \partial _{{\bar{t}}}|V^n|_1^2. \end{aligned}$$

We get (A.2). Similarly, we have for \(V^n\in {\mathcal {V}}_h\)

$$\begin{aligned} (\Delta _h^2({\partial }_{{\bar{t}}}V^n),{\bar{V}}^n)& = (\Delta _h({\partial }_{{\bar{t}}}V^n),\Delta _h{\bar{V}}^n) \\& = \displaystyle \frac{h_1h_2}{4\tau }\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}\Delta _h(V_{i,j}^{n+1}- V_{i,j}^{n-1})\cdot \Delta _h(V_{i,j}^{n+1}+ V_{i,j}^{n-1})\\& = \displaystyle \frac{1}{4\tau }(\Vert \Delta _hV^{n+1}\Vert ^2 -\Vert \Delta _hV^{n-1}\Vert ^2) \\& = \displaystyle \frac{1}{2} \partial _{{\bar{t}}}\Vert \Delta _h V^n\Vert ^2, \end{aligned}$$

and (A.3) follows. For any \(V^n, W^n \in {\mathcal {V}}_h,\) we have

$$\begin{aligned} ({\nabla }_h{\bar{V}}^n,{\bar{W}}^n)=-({\bar{V}}^n, {\nabla }_h{\bar{W}}^n), \end{aligned}$$

in particular, if \({\bar{V}}^n={\bar{W}}^n\), then

$$\begin{aligned} ({\nabla }_h{\bar{V}}^n,{\bar{V}}^n)=-({\bar{V}}^n, {\nabla }_h{\bar{V}}^n). \end{aligned}$$


$$\begin{aligned} ({\nabla }_h{\bar{V}}^n,{\bar{V}}^n)=0, \end{aligned}$$

we find (A.4). In view of difference properties and (2.5), we have

$$\begin{aligned} (V^n{\nabla }_h{\bar{V}}^n,{\bar{V}}^n)= ({\nabla }_h{\bar{V}}^n,V^n{\bar{V}}^n)= -({\bar{V}}^n, {\nabla }_h(V^n{\bar{V}}^n)). \end{aligned}$$

The above equality becomes

$$\begin{aligned} (\phi (V^n,\bar{V^n}),{\bar{V}}^n)=\displaystyle \frac{1}{3}(V^n{\nabla }_h{\bar{V}}^n,{\bar{V}}^n) +\displaystyle \frac{1}{3}({\nabla }_h(V^n{\bar{V}}^n),{\bar{V}}^n)=0. \end{aligned}$$

This completes the proof of the Lemma 1. \(\square\)

Lemma 2

For \(V^n\in {\mathcal {V}}_h,\) we have

$$\begin{aligned} \Vert \nabla _h V^n\Vert\le & |V^n|_1, \end{aligned}$$
$$\begin{aligned} \Vert \nabla _h V^n\Vert ^2\le & \Vert V^n\Vert .\Vert \Delta _h V^n\Vert , \end{aligned}$$
$$\begin{aligned} \Vert V^n\Vert _{\infty }^2\le & C \Vert V^n\Vert \left( \Vert \Delta _h V^n\Vert +\Vert V^n\Vert \right) . \end{aligned}$$


For \(V^n\in {\mathcal {V}}_h,\) we have

$$\begin{aligned} \Vert \nabla _h V^n\Vert ^2& = \Vert V_{{\hat{x}}}^n\Vert ^2+\Vert V_{{\hat{y}}}^n\Vert ^2\\& = h_1h_2\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}\left( \displaystyle \frac{V_{i+1,j}^{n}- V_{i-1,j}^{n}}{2h_1}\right) ^2 \\&+ h_1h_2\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}\left( \displaystyle \frac{V_{i,j+1}^{n}- V_{i,j-1}^{n}}{2h_2}\right) ^2 \\& = h_1h_2\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}\left( \displaystyle \frac{V_{i+1,j}^{n}- V_{i,j}^{n}}{2h_1}+\displaystyle \frac{V_{i,j}^{n}- V_{i-1,j}^{n}}{2h_1} \right) ^2 \\&+ h_1h_2\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}\left( \displaystyle \frac{V_{i,j+1}^{n}- V_{i,j}^{n}}{2h_2}+\displaystyle \frac{V_{i,j}^{n}- V_{i,j-1}^{n}}{2h_2} \right) ^2 \\\le & 2h_1h_2\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}\left( \displaystyle \frac{V_{i+1,j}^{n}- V_{i,j}^{n}}{2h_1}\right) ^2\\&+2h_1h_2\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}\left( \displaystyle \frac{V_{i,j}^{n}- V_{i-1,j}^{n}}{2h_1} \right) ^2 \\&+ 2h_1h_2\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}(\displaystyle \frac{V_{i,j+1}^{n}- V_{i,j}^{n}}{2h_2})^2\\&+2h_1h_2\displaystyle \sum _{i=1}^{M_1}\displaystyle \sum _{j=1}^{M_2}(\displaystyle \frac{V_{i,j}^{n}- V_{i,j-1}^{n}}{2h_2} )^2. \end{aligned}$$


$$\begin{aligned} \Vert \nabla _h V^n\Vert ^2\le \displaystyle \frac{1}{2}\Vert V_x^n\Vert ^2+\displaystyle \frac{1}{2}\Vert V_{{\bar{x}}}^n\Vert ^2+\displaystyle \frac{1}{2}\Vert V_y^n\Vert ^2+\displaystyle \frac{1}{2}\Vert V_{{\bar{y}}}^n\Vert ^2. \end{aligned}$$

From the properties of differences and periodic boundary, we obtain

$$\begin{aligned} \Vert V_x^n\Vert ^2=\Vert V_{{\bar{x}}}^n\Vert ^2\hbox { and }\Vert V_y^n\Vert ^2=\Vert V_{{\bar{y}}}^n\Vert ^2. \end{aligned}$$

This yields that

$$\begin{aligned} \Vert \nabla _h V^n\Vert ^2\le \Vert V_x^n\Vert ^2+\Vert V_y^n\Vert ^2=|V^n|_1^2, \end{aligned}$$

and (A.6) follows. By the discrete Green formula, we have for \(V^n\in {\mathcal {V}}_h\)

$$\begin{aligned} |V^n|_1^2=-(V^n, \Delta _hV^n)\le \Vert V^n\Vert \cdot \Vert \Delta _hV^n\Vert . \end{aligned}$$

The claimed inequality (A.7) follows from (A.6) immediately.

We can see the proof of the inequality (A.8) in [35]. \(\square\)

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Omrani, K., Ghiloufi, A. An efficient computational approach for two-dimensional variant of nonlinear-dispersive model of shallow water wave. Engineering with Computers (2020).

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  • Shallow water wave
  • Rosenau-RLW equation
  • Linearized difference scheme
  • Solvability
  • Stability
  • Convergence

Mathematics Subject Classification

  • 65M06
  • 65M12
  • 65M15