A cubic B-spline Galerkin approach for the numerical simulation of the GEW equation S .

The generalized equal width (GEW) wave equation is solved numerically by using lumped Galerkin approach with cubic B-spline functions. The proposed numerical scheme is tested by applying two test problems including single solitary wave and interaction of two solitary waves. In order to determine the performance of the algorithm, the error norms L2 and L∞ and the invariants I1, I2 and I3 are calculated. For the linear stability analysis of the numerical algorithm, von Neumann approach is used. As a result, the obtained findings show that the presented numerical scheme is preferable to some recent numerical methods.


Introduction
Firstly, Peregrine [1] presented the regularized long wave (RLW) equation 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.Later, Korteweg-de Vries (KdV) equation, which describes the long waves in non-linear dispersive systems, was introduced by Benjamin et al. [2].The equal width (EW) wave equation was used by Morrison et al. [3] as an alternative model to the RLW and the KdV equations.The EW equation is obtained by taking p = 1 in the generalized equal width (GEW) wave equation.So, the GEW equation is based upon the EW equation and is related to the generalized regularized long wave (GRLW) equation and the generalized Korteweg-de Vries (GKdV) equation.These general equations are nonlinear wave equations with (p + 1)th nonlinearity and have solitary solutions, which are pulse-like Raslan [7].The GEW equation is given by the following form: with the boundary and initial conditions If p = 2 in Eq.( 1), the obtained equation is known as the modified equal width (MEW) wave equation.The MEW equation has been studied by many researchers.A lumped Galerkin method was set up by Esen [17] with quadratic B-splines.Collocation approach was used by Saka [18] for the numerical solution of the MEW equation.A lumped Galerkin method and Petrov-Galerkin method based on cubic B-splines have been implemented to the MEW equation by Karakoc ¸and Geyikli [19,20].
In the literature, there are limited number of studies on the GEW equation.The exact solitary wave solutions of the generalized EW and the generalized EW-Burges equation were obtained by Hamdi et al. [5].Evans and Raslan [6], Raslan [7] presented the collocation method based on quadratic, cubic B-splines to get the numerical solution of the GEW equation.Petrov-Galerkin finite element method using a quadratic B-spline function as the trial function was investigated for solving the GEW equation by Roshan [8].The GEW equation was solved numerically using the meshless method based on a global collocation with standard types of radial basis functions (RBFs) by Panahipour [9].Taghizadeh et al. [10] have constructed the homogeneous balance method to obtain the exact travelling wave solutions of the GEW equation.
Galerkin finite element method based on B-spline functions, which is discussed here, has been used to obtain the numerical solution of nonlinear modeling problems by many authors.Gardner and Gardner [21] applied the Galerkin method with cubic B-splines to the RLW equation.Dogan [22] introduced the Galerkin's method using linear space finite elements to obtain the numerical results of the RLW equation.Dag et al. [23] solved the RLW equation using the quintic B-spline Galerkin approach.Saka and Dag [24], Kutluay and Uc ¸ar [25], Karakoc ¸et al. [26] have proposed the Galerkin method based on quartic, quadratic, cubic B-spline functions to acquire the numerical solutions of KdVB, Coupled KdV, MRLW equations, respectively.Lately, numerical solutions of the fractional diffusion and fractional diffusion-wave equations, an Improved Boussinesq type equation, a coupled mKdV equation have been obtained by means of quadratic B-spline Galerkin scheme [28,29,30].Also, B-splines have been used for applying the collocation method as an approximation function to get the numerical solution of Kawahara equation [27].
When we look at the numerical results of nonlinear modeling problems, the Galerkin approach is an accurate and efficient numerical technique.Besides, because of the second derivative to x in Eq.( 1), the approximate function must be at least second order (quadratic, cubic and so on).That's why , in this paper, we have implemented the lumped Galerkin method using cubic B-splines to the GEW equation.

A Lumped Galerkin Method
Let us consider the solution domain limited to a finite interval x ∈ [x m+1 , x m+2 ], 0 otherwise. ( Because each cubic B-spline ϕ m covers 4 intervals, each finite interval [x m , x m+1 ] is covered by 4 splines.The approximate solution U N (x, t) is written in terms of the cubic B-spline functions as in which the unknown δ j (t) are time-dependent quantities and they will be calculated by using the boundary and weighted residual conditions.Using the equality hη is converted into more easily workable interval [0, 1].In this case, the cubic B-splines (3) depending on variable η over the gap [0, 1] are reconstructed in the following form: ( Here we should state that all cubic B-spline functions except that ϕ m−1 (x), ϕ m (x), ϕ m+1 (x) and ϕ m+2 (x) are null over the finite element [0, 1].Therefore, approximation function (4) in terms of element parameters δ m−1 , δ m , δ m+1 , δ m+2 and B-spline element shape functions ϕ m−1 , ϕ m , ϕ m+1 , ϕ m+2 can be defined over the interval [0, 1] by Using B-splines (5) and trial function (6), we can write the nodal values of U, U ′ , U ′′ with respect to the time parameters δ m in the following form: where the superscript ′ and ′′ symbolize the first and second derivative to η, respectively.When applying the Galerkin's approach with weight function W (x) to Eq.( 1), we get the weak form of Eq.( 1) as follows: Implementing the change of variable x → η to integral (8) yields where Ů is taken to be a constant over an element to simplify the integral.Applying partial integration once to (9) forms where λ = ε Ů p h and β = µ h 2 .Substituting cubic B-splines (5) instead of the weight function W (x) and trial function (6) into integral equation (10) yield to the following form: in which δ e = (δ m−1 , δ m , δ m+1 , δ m+2 ) T and the dot states differentiation to t.This equation can be written in matrix form by The element matrices are explained as follows: ) p is By considering together contributions from all elements, the matrix equation ( 12) becomes where ) where

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A CUBIC B-SPLINE GALERKIN APPROACH FOR THE GEW EQUATION Applying the forward finite difference δ = δ n+1 −δ n ∆t and Crank-Nicolson approach δ = 1 2 (δ n + δ n+1 ) to equation ( 13), we can easily achieve the septa-diagonal matrix system Using the boundary conditions given by (2), the (N + 3) × (N + 3) system ( 14) is reduced to (N + 1) × (N + 1) septa-diagonal matrix system.This equation system can be solved by using Thomas algorithm.In this solution process, we need to two or three inner iterations δ n * = δ n + 1 2 (δ n − δ n−1 ) at each time step to minimize the effect of non-linearity.Eventually, we obtain the recurrence relationship between two time steps n and n + 1 as an ordinary member of the matrix system ( 14) where In order to start the iteration, the initial vector δ 0 must be computed by using the initial and boundary conditions.Because of this, using the relations at the knots together with a variant of the Thomas algorithm, the initial vector δ 0 can be easily calculated from the following matrix form

Stability analysis
For the linear stability analysis of the numerical algorithm, we use the Fourier method and assume that the quantity U p in the non-linear term U p U x of GEW equation is locally constant.Substituting the Fourier mode δ n j = g n e ijkh where k is mode number and h is the element size, into the scheme (15), this leads to the growth factor where The modulus of |g| is 1 which means that the scheme is unconditionally stable.

Numerical Examples and Results
The GEW equation has the solitary wave solution [6,7,8]

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where c is the the constant velocity of the wave traveling in the positive direction of the x-axis, x 0 is arbitrary constant.Moreover, the equation possesses the three conservation laws which correspond to mass, momentum and energy.The numerical algorithm is tested with single solitary wave and interaction of two solitary wave problems.In these two problems, to measure the performance of the numerical method, the L 2 and L ∞ error norms are computed by using the solitary wave solution in (18) and the following equalities: The changes of the invariants (19) are also observed to indicate the conservation properties of the numerical approach.

The Motion of Single Solitary Wave
For this problem, the five sets of parameters by taking different values of p, c and amplitude and the same values of h = 0.1, ∆t = 0.2, ε = 3, µ = 1, x 0 = 30, 0 ≤ x ≤ 80 is considered to coincide with papers [6,7,8].The numerical simulations are run from the time t = 0 to time t = 20.
In the first case, we choose the quantities p = 2, c = 1/32 and 1/2.Hence, the solitary wave has amplitude = 0.25 and 1, respectively.The calculated quantities of the invariants are presented in Tables I, II.As can be seen in Table I, three invariants are almost constant as the time increases.Table II shows that the changes of the invariants from their initial state are less than 2%, 3% and 3%, respectively.Also, we have found out that the quantity of the error norms L 2 and L ∞ is reasonably small, as expected.Secondly, if p = 3, c = 0.001 and c = 0.3, the solitary wave has amplitude = 0.15 and 1.The obtained results are given in Tables III, IV.It is observed from Table III that three invariants are nearly unchanged as the time processes.In Table IV, the changes of the invariants are less than 2%, 3% and 3%, respectively.In addition, The values of the error norms L 2 and L ∞ are adequately small.Table III.The invariants and the error norms for single solitary wave with p = 3, amplitude = 0.15, ∆t = 0.2, h = 0.1, ε = 3, µ = 1, 0 ≤ x ≤ 80. Finally, we take the parameters p = 4, c = 0.2.This leads to amplitude = 1.The obtained results are listed in Table V which clearly shows that the change of the invariants from their initial count are less than 2%.Also, we observed that the quantity of the error norms L 2 and L ∞ is sensibly small.The motion of a single solitary wave is plotted at different time levels t = 0, 10, 20 in Fig. 1, 2. It is understood from these figures that the numerical scheme performs the motion of propagation of a single solitary wave, which moves to the right at nearly unchanged speed and conserves its amplitude and shape with increasing time.
The comparison of our results with the ones obtained by collocation methods based on quadratic, cubic B-spline [6,7] and Petrov-Galerkin method [8] at t = 20 is given in Table VI.From this table, we can conclude that the values of three invariants are to be close to each other.The magnitude of our error norms is smaller than the ones given by [6,7] for p = 2, 3 and it is almost same with the paper [8] for p = 4.

The Interaction of Two Solitary Waves
In the second test problem, we use the initial condition which produces two positive solitary waves having different amplitudes of magnitudes 1 and 0.5 at the same direction, where c i and x i , i = 1, 2 are arbitrary constants.
Table VII.The invariants for interaction of two solitary waves with p = 2, c 1 = 0.5, c 2 = 0.125, Ours-Galerkin Pet.-Gal.[8] Ours-Galerkin Pet.-Gal.[ The interaction of two solitary waves is depicted at different time levels in Fig. 3, 4(a)-(d).In these figures, initially, the wave with larger amplitude is on the left of the second wave with smaller amplitude.In the progress of time, the large wave catches up with the smaller one and overlapping process occurs.In time, waves start to resume their original shapes.

Conclusion
In this paper, we have obtained the solitary-wave solutions of the GEW equation using lumped Galerkin method based on cubic B-spline functions.To prove the performance of numerical scheme, the error norms L 2 and L ∞ for single solitary wave and three invariants I 1 , I 2 and I 3 for two test problems have been calculated.These calculations represent that our error norms are adequately small and they are smaller than or too close to the ones in existing numerical results.The changes of the invariants are sufficiently small and the quantities of the invariants are consistent with those of Roshan.Also, the linearized numerical scheme is unconditionally stable.Finally, we can say that our numerical method can be reliably used to obtain the numerical solution of the GEW equation and similar type non-linear equations.