# A numerical investigation of the GRLW equation using lumped Galerkin approach with cubic B-spline

- Halil Zeybek
^{1}Email author and - S. Battal Gazi Karakoç
^{2}

**Received: **29 December 2015

**Accepted: **12 February 2016

**Published: **27 February 2016

## Abstract

In this work, we construct the lumped Galerkin approach based on cubic B-splines to obtain the numerical solution of the generalized regularized long wave equation. Applying the von Neumann approximation, it is shown that the linearized algorithm is unconditionally stable. The presented method is implemented to three test problems including single solitary wave, interaction of two solitary waves and development of an undular bore. To prove the performance of the numerical scheme, the error norms \(L_{2}\) and \({L_{\infty}}\) and the conservative quantities \({I_{1}}\), \({I_{2}}\) and \({I_{3}}\) are computed and the computational data are compared with the earlier works. In addition, the motion of solitary waves is described at different time levels.

### Keywords

GRLW equation Lumped Galerkin method Cubic B-spline Solitary waves Undular bore### Mathematics Subject Classification

41A15 65N30 76B25## Background

The generalized regularized long wave (GRLW) equation, which discussed here, is based upon the regularized long wave (RLW) equation. The RLW equation was firstly derived from long waves propagating in the positive *x*-direction as a model for small-amplitude long waves on the surface of water in a channel by Peregrine (1966, 1967). Benjamin et al. (1972) introduced the RLW equation as a reasonable alternative model to the more common Korteweg-de Vries (KdV) equation. The KdV equation describes the long waves with assumption of small wave amplitude and large wave length in non-linear dispersive and many other physical systems. Later, the equal width (EW) wave equation was used by Morrison et al. (1984) as an alternative model to the RLW equation. So, the GRLW equation is related to the generalized equal width (GEW) wave equation and the generalized Korteweg-de Vries (GKdV) equation. These general equations are nonlinear wave equations with \((p+1)\hbox {th}\) nonlinearity and have solitary wave solutions, which are pulse-like.

*t*and

*x*represent time and spatial differentiation, \(\varepsilon \) and

*p*is a positive integer, \(\mu \) is positive constant. The boundary and initial conditions are taken

*f*(

*x*) is a localized disturbance inside the interval [

*a*,

*b*] and it will be considered later. In the fluid problems,

*U*implies the vertical displacement of the water surface or similar physical quantity. In the plasma applications,

*U*is denoted as negative of the electrostatic potential. That’s why, the solitary wave solution of Eqs. (1), (2) and (3) helps us to understand the many physical phenomena with weak nonlinearity and dispersion waves such as nonlinear transverse waves in shallow water, ion-acoustic and magnetohydrodynamic waves in plasma and phonon packets in nonlinear crystals.

The RLW equation is obtained by taking \(p=1\) in GRLW equation (3). Up to now, many numerical including finite elements and analytical solution techniques have been presented on the RLW equation. The RLW equation was investigated with the growth of an undular bore by Peregrine (1966). Morrison et al. (1984) proposed the approximate analytical technique for the scattering of solitary waves of the RLW equation. Galerkin approach with linear, quadratic and quintic B-spline was used by Doğan (2002), Gardner et al. (1995) and Dağ et al. (2006). Collocation method was set up by Raslan (2001) and Saka et al. (2011) with quadratic and both sextic and septic B-splines functions. Esen and Kutluay (2006) obtained the numerical solution of the RLW equation with lumped Galerkin method using quadratic B-spline. Galerkin method with extrapolation techniques has been implemented to the RLW equation by Mei and Chen (2012). Later on, the RLW equation has been solved numerically by using von Neumann technique based on parametric quintic splines (Lin 2014).

If \(p=2\) in Eq. (3), the obtained equation is called as the modified regularized long wave (MRLW) equation. Finite element methods based on quintic, cubic and septic collocation were used for obtaining the numerical solution of the MRLW equation by Gardner et al. (1997), Khalifa et al. (2008) and Karakoç et al. (2014). Collocation method based on quintic B-spline functions with Rubin and Graves linearization technique was investigated for solving the MRLW equation by Karakoç et al. (2013). The MRLW equation was solved numerically by Ali (2009) using mesh free collocation method. Galerkin approach with cubic B-spline has been applied to MRLW equation by Karakoç et al. (2015).

When we consider the GRLW equation discussed here, there are some exact and numerical solution techniques on its. Hamdi et al. (2004) presented the exact solution technique. Numerical methods based on decomposition scheme, finite difference scheme and element free kp-Ritz were introduced for GRLW equation by Kaya (2004), EL-Danaf et al. (2014) and Guo et al. (2014). An approximate quasilinearization scheme was used to solve the GRLW equation with initial condition on the formation of undular bore by Ramos (2007). Roshan (2012) and Mohammadi (2015) have got the numerical results of the GRLW equation using finite element method based on Petrov Galerkin and exponential B-spline collocation. Also, Galerkin and lumped Galerkin method used here have been implemented to the EW, KdVB, Coupled KdV and MEW equations by Doğan (2005), Saka and Dağ (2009), Kutluay and Uçar (2013) and Esen (2006).

Inspired by the results of the applied numerical methods to similar type equations, we can say that lumped Galerkin approach is an accurate and efficient numerical technique. So, in this work, we have constructed the lumped Galerkin approach with cubic B-splines to get the numerical results of the GRLW equation.

## A lumped Galerkin method

*a*,

*b*] is divided into

*N*equal subinterval by the points \(x_{m}\) such that \( a=x_{0}<x_{1}\ldots <x_{N}=b\) and length \(h=\frac{b-a}{N}=(x_{m+1}-x_{m})\). Prenter (1975) described the cubic B-spline functions \(\phi _{m}(x)\), (

*m*= \(-1(1)\) \(N+1\)), at the nodes \(x_{m}\) which form a basis over the interval [

*a*,

*b*] by

*W*(

*x*) to Eq. (3), the weak form of Eq. (3) is obtained as

*W*(

*x*) and trial function (8) into integral equation (12) forms

*t*, which can be written in matrix form by

*A*,

*B*,

*C*and \(\lambda D\) are septa-diagonal matrices and their line of

*m*is

*A*,

*B*,

*C*and

*D*has seven elements, the system (16) comprises of the diagonal matrix with seven columns element (known as septa-diagonal matrix). The septa-diagonal matrix system can be solved by using Thomas algorithm (see subsection ). In this solution procedure, we need to two or three inner iterations \(\delta ^{n*}=\delta ^{n}+\frac{1}{2}\left( \delta ^{n}-\delta ^{n-1}\right) \) at each time step to minimize the non-linearity. After all of these processes, we can easily achieve the recurrence relationship between two time steps

*n*and \(n+1\) which is an ordinary member of the matrix system (16)

### The solution of septa-diagonal matrix system with Thomas algorithm

### Stability analysis

*k*is mode number,

*h*is the element size and \(i = {\sqrt{-1}}\), into the scheme (17), which produces the following equality

*g*| is 1, so the linearized scheme is unconditionally stable.

## Numerical examples and results

*x*-axis, \(x_{0}\) is arbitrary constant.

### The motion of single solitary wave

For this problem, we use the initial condition obtained by taking \(t=0\) in Eq. (22). To coincide with papers Dağ et al. (2006), Gardner et al. (1997), Khalifa et al. (2008), Ali (2009), Karakoç et al. (2013), Roshan (2012) and Mohammadi (2015), the same values of \(\mu = 1\), \({x_{0}}=40\), \({x\in} \left[ 0,100 \right] \) and different values of *p*, *c*, *h*, \({\Delta {t}}\) are considered. The numerical computations are run from the time \(t=0\) to time \(t=10\) or \( t=20\).

Firstly, we choose the quantities \(p=2\), \(c=1\), \(h=0.2\), \({\Delta {t}} = 0.025\) and \(p=2\), \(c=0.3\), \(h=0.1\), \({\Delta {t}} = 0.01\). These values yield the \( amplitude=1\) and \(amplitude=0.54772\). The obtained results are given in Tables 1 and 2. It is observed from Table 1 that the changes of the invariants are less than 0.04, 0.05 and 0.05 %, respectively. In Table 2, three invariants are nearly unchanged as the time processes. Moreover, The values of the error norms \( {L_{2}}\) and \({L_{\infty}}\) are adequately small.

Invariants and errors for single solitary wave with \(p = 2, c = 1, h = 0.2,{\Delta{t}} = 0.025,\mu = 1,x\in \left[ 0,100\right] \)

Time | \({I_{1}}\) | \({{I_{2}}}\) | \({{I_{3}}}\) | \({L_{2}}\times 10^{3}\) | \({L_{\infty}}\times 10^{3}\) |
---|---|---|---|---|---|

0 | 4.4428661 | 3.2998133 | 1.4142140 | 0.00000000 | 0.00000000 |

2 | 4.4429408 | 3.2999387 | 1.4143308 | 1.95082039 | 1.19160336 |

4 | 4.4430058 | 3.3000340 | 1.4144250 | 2.36484347 | 1.22370847 |

6 | 4.4430683 | 3.3001243 | 1.4145151 | 2.45181423 | 1.20000405 |

8 | 4.4431302 | 3.3002134 | 1.4146042 | 2.45030808 | 1.15204959 |

10 | 4.4431919 | 3.3003022 | 1.4146930 | 2.41750291 | 1.08099621 |

Invariants and errors for single solitary wave with \( p=2,c=0.3,h=0.1,\Delta t=0.01,\mu =1,x\in \left[ 0,100\right] \)

Time | \({I_{1}}\) | \({I_{2}}\) | \({I_{3}}\) | \({L_{2}}\times 10^{4}\) | \({L_{\infty}}\times10^{4}\) |
---|---|---|---|---|---|

0 | 3.5820205 | 1.3450941 | 0.1537283 | 0.00000000 | 0.00000000 |

4 | 3.5820206 | 1.3450942 | 0.1537284 | 0.87664666 | 0.42835220 |

8 | 3.5820207 | 1.3450943 | 0.1537284 | 1.09331524 | 0.42259060 |

12 | 3.5820207 | 1.3450943 | 0.1537284 | 1.16711699 | 0.42542846 |

16 | 3.5820207 | 1.3450944 | 0.1537284 | 1.20368923 | 0.43881496 |

20 | 3.5820206 | 1.3450944 | 0.1537284 | 1.22736382 | 0.44722941 |

Invariants and errors for single solitary wave with \( p=3,c=1.2,h=0.1,{\Delta{t}}=0.025,\mu =1,x\in \left[ 0,100\right] \)

Time | \(I_{1}\) | \({{I_{2}}}\) | \({{I_{3}}}\) | \({L_{2}}\times 10^{3}\) | \({L_{\infty}}\times 10^{3}\) |
---|---|---|---|---|---|

0 | 3.7971850 | 2.8812503 | 0.9729681 | 0.00000000 | 0.00000000 |

2 | 3.7980891 | 2.8826274 | 0.9747778 | 6.37523435 | 4.16206480 |

4 | 3.7989816 | 2.8839827 | 0.9760069 | 10.53160077 | 6.58017074 |

6 | 3.7998750 | 2.8853393 | 0.9771207 | 13.02367954 | 8.10106559 |

8 | 3.8007710 | 2.8867002 | 0.9782095 | 13.93740889 | 8.73017950 |

10 | 3.8016702 | 2.8880662 | 0.9792942 | 13.29108053 | 8.47810737 |

Invariants and errors for single solitary wave with \( p=3,c=0.3,h=0.1,{\Delta{t}}=0.01,\mu =1,x\in \left[ 0,100\right] \)

Time | \(I_{1}\) | \({I_{2}}\) | \({I_{3}}\) | \({L_{2}}\times 10^{4}\) | \({L_{\infty}}\times 10^{4}\) |
---|---|---|---|---|---|

0 | 3.6776069 | 1.5657603 | 0.2268463 | 0.00000000 | 0.00000000 |

2 | 3.6776071 | 1.5657606 | 0.2268544 | 1.18720589 | 0.73102952 |

4 | 3.6776072 | 1.5657607 | 0.2268573 | 1.60659681 | 0.88913800 |

6 | 3.6776072 | 1.5657607 | 0.2268575 | 1.76861454 | 0.81537826 |

8 | 3.6776072 | 1.5657607 | 0.2268575 | 1.85663605 | 0.75460192 |

10 | 3.6776072 | 1.5657608 | 0.2268574 | 1.91332225 | 0.77992648 |

Invariants and errors for single solitary wave with \( p=4,c=4/3,h=0.1,{\Delta{t}}=0.01,\mu =1,x\in \left[ 0,100\right] \)

Time | \({I_{1}}\) | \({I_{2}}\) | \({I_{3}}\) | \({L_{2}}\times 10^{3}\) | \({L_{\infty}}\times 10^{3}\) |
---|---|---|---|---|---|

0 | 3.4687090 | 2.6716914 | 0.7292045 | 0.00000000 | 0.00000000 |

2 | 3.4690660 | 2.6722659 | 0.7305244 | 2.71272493 | 1.97322350 |

4 | 3.4694090 | 2.6728105 | 0.7309610 | 3.80159123 | 2.65902173 |

6 | 3.4697519 | 2.6733547 | 0.7313161 | 3.84205549 | 2.71392029 |

8 | 3.4700954 | 2.6738997 | 0.7316538 | 2.88903866 | 2.11361885 |

10 | 3.4704395 | 2.6744459 | 0.7319875 | 1.51139451 | 0.85758574 |

Invariants and errors for single solitary wave with \( p=4,c=0.3,h=0.1,{\Delta{t}}=0.01,\mu =1,x\in \left[ 0,100\right] \)

Time | \({I_{1}}\) | \({I_{2}}\) | \({I_{3}}\) | \({L_{2}}\times 10^{4}\) | \({L_{\infty}}\times 10^{4}\) |
---|---|---|---|---|---|

0 | 3.7592865 | 1.7300236 | 0.2894191 | 0.00000000 | 0.00000000 |

2 | 3.7592871 | 1.7300246 | 0.2894498 | 1.91721709 | 1.20079691 |

4 | 3.7592873 | 1.7300248 | 0.2894559 | 2.45184081 | 1.44560973 |

6 | 3.7592874 | 1.7300249 | 0.2894566 | 2.70531310 | 1.21535724 |

8 | 3.7592874 | 1.7300250 | 0.2894569 | 2.90077790 | 1.31685490 |

10 | 3.7592875 | 1.7300251 | 0.2894570 | 3.08940237 | 1.44471990 |

*p*and

*c*. In addition, the motion of single solitary wave is displayed at different times and the values of

*p*in Fig. 1. From this figure, we can see that the solitary wave moves to the right at constant velocity and remains its shape and amplitude. When the values of

*p*are increased, the peak position of single solitary wave rises.

Errors for single solitary wave with \(h=0.1,{\Delta{t}}=0.01,\mu =1,x\in \left[ 0,100\right] \)

p = 2 | p = 3 | p = 4 | p = 6 | p = 8 | p = 10 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

c | 0.03 | 0.1 | 0.03 | 0.1 | 0.03 | 0.1 | 0.03 | 0.1 | 0.03 | 0.1 | 0.03 | 0.1 |

amp | 0.17 | 0.31 | 0.29 | 0.43 | 0.38 | 0.52 | 0.52 | 0.63 | 0.60 | 0.70 | 0.66 | 0.75 |

Time | ||||||||||||

\({L_{2}}\times 10^{4}\) | ||||||||||||

5 | 4.36 | 0.16 | 5.84 | 0.37 | 6.89 | 0.65 | 8.26 | 1.44 | 9.12 | 2.76 | 9.71 | 5.09 |

10 | 5.15 | 0.27 | 6.91 | 0.52 | 8.15 | 0.88 | 9.78 | 2.24 | 10.80 | 5.61 | 11.53 | 13.26 |

15 | 5.28 | 0.36 | 7.08 | 0.63 | 8.35 | 1.08 | 10.02 | 3.25 | 11.08 | 9.92 | 11.91 | 27.67 |

20 | 5.54 | 0.44 | 7.43 | 0.74 | 8.77 | 1.29 | 10.53 | 4.51 | 11.67 | 15.92 | 12.66 | 51.36 |

\({L_{\infty}}\times 10^{4}\) | ||||||||||||

5 | 2.21 | 0.09 | 2.96 | 0.21 | 3.49 | 0.36 | 4.18 | 0.82 | 4.61 | 1.68 | 4.90 | 3.20 |

10 | 2.11 | 0.13 | 2.83 | 0.25 | 3.33 | 0.43 | 4.00 | 1.18 | 4.41 | 3.09 | 4.68 | 7.34 |

15 | 2.01 | 0.16 | 2.69 | 0.29 | 3.18 | 0.51 | 3.81 | 1.66 | 4.20 | 5.12 | 4.46 | 14.39 |

20 | 4.16 | 0.19 | 5.57 | 0.34 | 6.58 | 0.61 | 7.88 | 2.22 | 8.69 | 7.88 | 9.23 | 25.82 |

Comprasions of result for the single solitary wave with \(\mu =1,x\in \left[ 0,100\right] \)

Methods | \({L_{2}}\times 10^{3}\) | \({L_{\infty}}\times 10^{3}\) | \({I_{1}}\) | \( {I_{2}}\) | \({I_{3}}\) | |
---|---|---|---|---|---|---|

\(p=2\) | CBSC-CN (Gardner et al. 1995) | 16.3900 | 9.2400 | 4.4420 | 3.2990 | 1.4130 |

\(c=1\) | CBSC+PA-CN (Gardner et al. 1995) | 20.3000 | 11.2000 | 4.4400 | 3.2960 | 1.4110 |

\(h=0.2\) | CBSC (Khalifa et al. 2008) | 9.3019 | 5.4371 | 4.4428 | 3.2998 | 1.4142 |

\({\Delta{t}}=0.025\) | MFC (Ali 2009) | 3.9140 | 2.0190 | 4.4428 | 3.2997 | 1.4141 |

\(t=10\) | QBSPG (Roshan 2012) | 3.0053 | 1.6874 | 4.4428 | 3.2998 | 1.4141 |

QBSC (Karakoç et al. 2013) | 2.4155 | 1.0797 | 4.4431 | 3.3003 | 1.4146 | |

EBSC (Mohammadi 2015) | 2.3909 | 1.0647 | 4.4428 | 3.2998 | 1.4142 | |

Ours-CBSG | 2.4175 | 1.0809 | 4.4431 | 3.3003 | 1.4146 | |

QBSPG (Roshan 2012) | ||||||

\(p=3\) | t = 1 | 0.0101 | 0.0080 | 3.6775 | 1.5657 | 0.2268 |

\(c=0.3\) | t = 5 | 0.0409 | 0.0238 | 3.6775 | 1.5657 | 0.2268 |

\(h=0.1\) | t = 10 | 0.0719 | 0.0377 | 3.6775 | 1.5657 | 0.2268 |

Ours-CBSG | ||||||

\({\Delta{t}}=0.01\) | t = 1 | 0.0706 | 0.0514 | 3.6776 | 1.5657 | 0.2268 |

t = 5 | 0.1702 | 0.0876 | 3.6776 | 1.5657 | 0.2268 | |

t = 10 | 0.1913 | 0.0779 | 3.6776 | 1.5657 | 0.2268 | |

QBSPG (Roshan 2012) | ||||||

\(p=4\) | t = 1 | 0.0158 | 0.0138 | 3.7592 | 1.7299 | 0.2894 |

\(c=0.3\) | t = 5 | 0.0542 | 0.0382 | 3.7592 | 1.7299 | 0.2894 |

\(h=0.1\) | t = 10 | 0.1225 | 0.0662 | 3.7592 | 1.7299 | 0.2894 |

Ours-CBSG | ||||||

\({\Delta{t}}=0.01\) | t = 1 | 0.1222 | 0.0983 | 3.7592 | 1.7300 | 0.2894 |

t = 5 | 0.2591 | 0.1357 | 3.7592 | 1.7300 | 0.2894 | |

t = 10 | 0.3089 | 0.1444 | 3.7592 | 1.7300 | 0.2894 |

### The interaction of two solitary waves

Invariants for interaction of two solitary waves with \( p=2,c_{1}=4,c_{2}=1,x_{1}=25,x_{2}=55,h=0.2,{\Delta{t}}=0.025,\mu =1,x\in \left[ 0,250\right] \)

Time | \({I_{1}}\) | \({I_{2}}\) | \({I_{3}}\) | |||
---|---|---|---|---|---|---|

Ours-CBSG | QBSPG (Roshan 2012) | Ours-CBSG | QBSPG (Roshan 2012) | Ours-CBSG | QBSPG (Roshan 2012) | |

0 | 11.4676 | 11.4677 | 14.6290 | 14.6286 | 22.8804 | 22.8788 |

4 | 11.4674 | 11.4677 | 14.6287 | 14.6292 | 22.8783 | 22.8811 |

8 | 11.4685 | 11.4677 | 14.6360 | 14.6229 | 22.9020 | 22.8798 |

12 | 11.4663 | 11.4677 | 14.6257 | 14.6299 | 22.8717 | 22.8803 |

16 | 11.4664 | 11.4677 | 14.6260 | 14.6295 | 22.8686 | 22.8805 |

20 | 11.4662 | 11.4677 | 14.6253 | 14.6299 | 22.8650 | 22.8806 |

Invariants for interaction of two solitary waves with \(p = 3\) and 4

Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

p=3 | |||||||

\({I_{1}}\) | 9.6907 | 9.6907 | 9.6906 | 9.6917 | 9.6898 | 9.6898 | 9.6901 |

\({I_{2}}\) | 12.9443 | 12.9443 | 12.9440 | 12.9489 | 12.9418 | 12.9420 | 12.9426 |

\({I_{3}}\) | 17.0187 | 17.0311 | 17.0324 | 18.0050 | 16.9849 | 16.9222 | 16.9557 |

p = 4 | |||||||

\({I_{1}}\) | 8.8342 | 8.7559 | 8.7089 | 8.6774 | 8.6518 | 8.6322 | 8.6134 |

\({I_{2}}\) | 12.1707 | 11.9304 | 11.7871 | 11.6932 | 11.6179 | 11.5560 | 11.4992 |

\({I_{3}}\) | 14.0296 | 13.3472 | 12.9204 | 13.2047 | 12.1972 | 12.0924 | 11.9640 |

### The development of an undular bore

*x*. Later, undulations take the peak position and disappear.

Invariants for development of an undular bore

Time | \({I_{1}}\) | \({I_{2}}\) | \({I_{3}}\) | ||||||
---|---|---|---|---|---|---|---|---|---|

p = 2 | p = 3 | p = 4 | p = 2 | p = 3 | p = 4 | p = 2 | p = 3 | p = 4 | |

Our results for \({U_{0}}=0.1,x_{0}=0,d=5,\mu =1/6,h=0.1,{\Delta{t}}=0.1,x\in \left[ -36,300\right] \) | |||||||||

0 | 3.5949 | 3.5949 | 3.5949 | 0.3344 | 0.3344 | 0.3344 | 0.0031 | 0.0031 | 0.0031 |

50 | 3.6051 | 3.6050 | 3.6049 | 0.3348 | 0.3350 | 0.3350 | 0.0019 | 0.0016 | 0.0015 |

100 | 3.6051 | 3.6050 | 3.6050 | 0.3348 | 0.3350 | 0.3350 | 0.0018 | 0.0016 | 0.0015 |

150 | 3.6050 | 3.6050 | 3.6049 | 0.3350 | 0.3349 | 0.3350 | 0.0017 | 0.0016 | 0.0015 |

200 | 3.6050 | 3.6050 | 3.6049 | 0.3354 | 0.3349 | 0.3350 | 0.0012 | 0.0016 | 0.0015 |

Time | \({I_{1}}\) | \({I_{2}}\) | \({I_{3}}\) | ||||||
---|---|---|---|---|---|---|---|---|---|

p = 2 | p = 2 | p = 2 | |||||||

QBSC[28] results for \({U_{0}}=0.1,d=5,\mu =3/2,h=0.2,{\Delta{t}}=0.1,x\in \left[ 0,250\right] \) | |||||||||

0 | 4.0000 | 0.3759 | 0.0025 | ||||||

50 | 4.8507 | 0.4620 | 0.0034 | ||||||

100 | 5.7016 | 0.5480 | 0.0042 | ||||||

150 | 6.5531 | 0.6341 | 0.0051 | ||||||

200 | 7.4055 | 0.7204 | 0.0060 |

## Conclusion

The solitary-wave solutions of the GRLW equation have been successfully obtained by using lumped Galerkin method based on cubic B-spline functions. Also, the linearized scheme has been found to be unconditonally stable. The error norms \({L_{2}}\), \({L_{\infty}}\) and three conservative quantities \({I_{1}}\), \({I_{2}}\) and \({I_{3}}\) have been computed for single solitary wave, interaction of two solitary waves and development of an undular bore. These computations demonstrate that our error norms are as small as required and they are smaller than the most of existing numerical calculations or too close to the best result in literature. The numerical algorithm conserves the properties related to mass, momentum and energy and the numerical values of them have been found to be in good agreement with earlier studies. In addition, the profiles of the solitary wave are similar to those of references. As a result, we can say that lumped Galerkin method is more practical, accurate and productive numerical approximation technique for GRLW equation and it can be reliably used to solve the similar type non-linear problems.

## Declarations

### Authors’ contributions

The authors worked with the consultation of each other to check and test programs, to obtain the results and prepared this work together. Both authors checked, corrected the final manuscript. Both authors read and approved the final manuscript.

**Acknowledgements**

The authors would like to express their sincere thanks to BioMed Central for contribution to this work and the reviewers for their careful reading, valuable comments and suggestions.

**Competing interests**

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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