Numerical solutions of the modified equal width wave equation are obtained by using the multigrid method and finite difference method. The motion of a single solitary wave, interaction of two solitary waves and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Using error norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other methods.
A large system of equations comes out from discretization of the domain of partial differential equations into a collection of points and the optimal method for solving these problems is multigrid method, see [1] - [4] .
The modified equal width wave (MEW) equation introduced by Morrison et al. [5] is used as a model equation to describe the nonlinear dispersive waves. Gardner and Gardner [6] [7] solved the EW equation with the Galerkin’s method using cubic B-splines as a trial and test function. The MEW equation was similar with the modified regularized long wave (MRLW) equation [8] and modified Korteweg-de Vries (MKdV) equation [9] . All the modified equations are nonlinear wave equations with cubic nonlinearities and all of them have solitary wave solutions, which are wave packets or pulses. These waves propagate in non-linear media by keeping wave forms and velocity even after interaction occurs.
Several solutions for MEW had been proposed in [10] - [22] . In Geyikli and Battal Gazi Karakoc [10] [11] , the solutions are based on septic B-spline finite elements and Petrov-Galerkin finite element method with weight functions quadratic and element shape functions which are cubic B-splines. Esen [12] [13] solved the MEW equation by applying a lumped Galerkin method based on quadratic B-spline finite elements. Saka [14] proposed algorithms for the numerical solution of the MEW equation using quintic B-spline collocation method. Zaki [15] considered the solitary wave interactions for the MEW equation by collocation method using quintic B-spline finite elements and obtained the numerical solution of the EW equation by using least-squares method [16] . Wazwaz [17] investigated the MEW equation and two of its variants by the tanh and the sine-cosine methods. A solution based on a collocation method incorporated cubic B-splines is investigated by Saka and Dağ [18] . Lu [19] presented a variational iteration method to solve the MEW equation. Evans and Raslan [20] studied the generalized EW equation by using collocation method based on quadratic B-splines to obtain the numerical solutions of a single solitary waves and the birth of solitons. Esen and Kutluay [21] studied a linearized implicit finite difference method in solving the MEW equation. Battal Gazi Karakoc and Geyikli [22] solved the MEW equation by a lumped Galerkin method using cubic B-spline finite elements.
An outline of this paper is as follows: We begin in Section 2 by reviewing the analytical solution of the MEW equation. In Section 3, we derive a new numerical method based on the multigrid technique and finite difference method for obtaining the numerical solution of MEW equation. Finally, in Section 4, we introduce the numerical results for solving the MEW equation through some well known standard problems.
2. The Analytical Solution
The modified equal width wave equation which is as a model for non-linear dispersive waves, considered here has the normalized form [5]
with the physical boundary conditions as, where t is time and x is the space coordinate, is a positive parameter. For this study boundary conditions are chosen
and the initial condition as
where f is a localized disturbance inside the considered interval.
The exact solution of equation (1) can be written in the form [15]
which represents the motion of a single solitary wave with amplitude A, where the wave velocity and. The initial condition is given by
For the MEW equation, it is important to discuss the following three invariant conditions given in [15] , which, respectively, correspond to conversation of mass, momentum, and energy. The analytical values of the invariants are
3. Numerical Method
The basic idea of multigrid techniques is illustrated by Brandt [1] . In this section we apply this method for initial boundary value problem, except that, the upper boundary conditions change with time, in which the initial condition is for. Dividing the interval of time to K parts, we obtain the solutions of the partial differential equation at time t1 and use these solutions as initial values for the next level, and for the other, we obtain the solutions at time T. The numbers of points in a coarse grid for this domain are two points.
We apply the full multigrid algorithm for the MRLW equation. Assuming the initial condition and the solution, has the usual partition with a space step size and a time step size ().
We start handling the non-linear term by expressing in the form. The back-time and centre-
space difference for Equation (1) is
where, for a set grids
Step 1:
Step 2: Starting from in the coarse grid, we can calculate the approximate value at two points using Equation (5) leading to:
The right hand side for equation (7) can be computed using the initial and boundary conditions.
Step 3: Interpolating the grid functions from the coarse grid to fine grid using linear interpolation, in which
that can be written explicitly as:
Step 4: Doing relaxation sweep on using the point relaxation
Step 5: Computing the residuals on and inject them into using full weighting restriction to get as:
Step 6: Computing an approximate solution of error.
Step 7: Interpolating the solution of error onto, and adding it to which is the approximate value of u on the fine grid with.
By taking this solution on coarse grid and repeating steps 3-7, we obtain the approximate values of u on the grid with and so the final value is the solution at the time level.
Step 8:, go to step 2 (lead to the solution at higher time level as needed).
4. Numerical Results
In this section, numerical solutions of MRLW equation are obtained for standard problems as: the motion of single solitary wave, interaction of two solitary waves and development of Maxwellian initial condition into solitary waves. For the MEW equation, it is important to discuss the following three invariant conditions given in [15] , which respectively correspond to conversation of mass, momentum and energy:
The accuracy of the method is measured by both the error norm
and the error norm
to show how good the numerical results in comparison with the exact results.
4.1. The Motion of Single Solitary Wave
Consider Equation (1) with boundary conditions (2) and the initial condition (4). For a comparison with earlier studies [13] [19] [21] [22] we take the parameters and over the interval [0, 80]. To find the error norms, and the numerical invariants and at various times we use the numerical solutions by applying the multigrid method up to. As reported in Table 1, the error norms, are found to be small enough, and the computed values of invariants are in good agreement with their analytical values Table 2 shows a comparison of the values of the invariants and error norms obtained by the present method with those obtained by other methods [13] [19] [21] [22] . It is clearly seen from Table 2 that the error norms obtained by the present method are smaller than the other methods.
4.2. Interaction of Two Solitary Waves
Consider the interaction of two positive solitary waves as a second problem. For this problem, the initial condition is given by:
For the computational discussion, firstly we use parameters and over the range to coincide with those used in [22] .
In [20] the analytic invariants are, ,. The experiment is run from to and values of the invariant quantities and are listed in Table 3.
Table 3 shows a comparison of the values of the invariants obtained by present method with those obtained in
Invariants and error norms for single solitary wave when<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/14-7403151x86.png" xlink:type="simple"/></inline-formula>
0
0.7853966199
0.1666662968
0.005208333331
0.000000000
0.000000
2
0.7853966246
0.1666660511
0.005208317956
0.0518705479
0.05440
4
0.7853966176
0.1666658044
0.005208302547
0.1038794545
0.10890
6
0.7853966097
0.1666655554
0.005208286962
0.1560469898
0.16359
8
0.7853966066
0.1666653078
0.005208271505
0.2080329043
0.21810
10
0.7853966012
0.1666650571
0.005208255823
0.2601313073
0.27283
12
0.7853965918
0.1666648091
0.005208240334
0.3122731279
0.32747
14
0.7853965793
0.1666645594
0.005208224692
0.3643751855
0.38216
16
0.7853965769
0.1666643124
0.005208209260
0.4164201991
0.43656
18
0.7853965785
0.1666640667
0.005208193877
0.4684782742
0.49095
20
0.7853965668
0.1666638167
0.005208178255
0.5208044265
0.54566
Comparison of errors and invariants for single solitary wave at<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/14-7403151x93.png" xlink:type="simple"/></inline-formula>
Method
Analytical
0.7853982
0.1666667
0.0052083
0
0
Present
0.7853966
0.1666638
0.0052082
0.520804
0.54566
[13]
0.7853898
0.1667614
0.0052082
7.969400
4.65523
[19]
0.7849545
0.1664765
0.0051995
29.05166
24.98925
[21]
0.7853977
0.1664735
0.0052083
26.92812
25.69972
[22]
0.7853967
0.1666663
0.0052083
8.009800
4.606180
Comparison of invariants for the interaction of two solitary waves with results from [<xref ref-type="bibr" rid="scirp.67675-ref22">22</xref>] (<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/14-7403151x100.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/14-7403151x100.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/14-7403151x99.png" xlink:type="simple"/></inline-formula>)
Present method
[22]
0
4.712379141
3.333328364
1.416669724
4.7123732
3.3333253
1.4166643
5
4.712378542
3.333075164
1.416419304
4.7123861
3.3333482
1.4166852
10
4.712378533
3.332822094
1.416169046
4.7123959
3.3333621
1.4166982
15
4.712378539
3.332569179
1.415918945
4.7124065
3.3333785
1.4167141
20
4.712378504
3.332316280
1.415668885
4.7124249
3.3334164
1.4167521
25
4.712378509
3.332063538
1.415418955
4.7124899
3.3335832
1.4169238
30
4.712378541
3.331810944
1.415169189
4.7127643
3.3333557
1.4177617
35
4.712378593
3.331558498
1.414919571
4.7130474
3.3352500
1.4188849
40
4.712378583
3.331306069
1.414669976
4.7124881
3.3336316
1.4171690
45
4.712378540
3.331053726
1.414420484
4.7123002
3.3331878
1.4167580
50
4.712378546
3.330801521
1.414171139
4.7122479
3.3330923
1.4167142
55
4.712378563
3.330632678
1.413975397
4.7122576
3.3331149
1.4167237
[22] . It is seen that the numerical values of the invariants remain almost constant during the computer run.
Finally, we have studied the interaction of two solitary waves with the following parameters and in the range [0,150].
The analytical invariants can be found as in [22] , ,. The experiment is run from t = 0 to t = 55 and values of the invariant quantities and are listed in Table 4.
4.3. The Maxwellian Initial Condition
Last study, we consider the numerical solution of the equation (1) with the Maxwellian initial condition
and the boundary conditions
Invariants for the interaction of two solitary waves (<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/14-7403151x118.png" xlink:type="simple"/></inline-formula>)
0
−3.141588324
13.33240988
22.66661773
5
−3.141587221
13.31632255
22.60298870
10
−3.141587293
13.30034113
22.53980538
15
−3.141587369
13.28446423
22.47706277
20
−3.141587465
13.26869077
22.41475674
25
−3.141587571
13.25301907
22.35288150
30
−3.141587642
13.23744806
22.29143237
35
−3.141587711
13.22197615
22.23040435
40
−3.141587744
13.20660137
22.16979117
45
−3.141587842
13.19132246
22.10958842
50
−3.141587954
13.17613770
22.04979002
55
−3.141587989
13.15897649
22.01765488
Invariants of MEW equation using the Maxwelliancondition<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/14-7403151x123.png" xlink:type="simple"/></inline-formula>
t
0
1.772450389
2.507031350
0.8862269258
1.772450389
2.507031350
0.8862269258
3
1.772450324
2.506562241
0.8859617965
1.772450391
2.503301389
0.8812594008
6
1
1.772450355
2.506093335
0.8856969273
0.05
1.772450389
2.508930167
0.8763574457
9
1.772450370
2.505624532
0.8854322207
1.772450398
2.523558486
0.874805062
12
1.772450368
2.505155693
0.8851675910
1.772450397
2.546856704
0.8665905426
0
1.772450389
2.507031350
0.8862269258
1.772450389
2.507031350
0.8862269258
3
1.772450359
2.505856717
0.8855663594
1.772450391
2.505380411
0.8790965969
6
0.5
1.772450306
2.504698981
0.8849067496
0.02
1.772450395
2.523617080
0.8721185562
9
1.772450294
2.503557663
0.882481160
1.772450301
2.561417967
0.8651660441
12
1.772450275
2.502431939
0.8835903023
1.772450306
2.587705888
0.8581212805
0
1.772450389
2.507031350
0.8862269258
1.772450389
2.507031350
0.8862269258
3
1.772450382
2.503247004
0.8830002438
1.772450388
2.509358370
0.8771798047
6
0.1
1.772450373
2.503178816
0.8797994434
0.005
1.772450389
2.545688708
0.8684172340
9
1.772450375
2.506705762
0.8766134367
1.772450390
2. 619139180
0.8596519203
12
1.772450380
2.513694005
0.8734315977
1.772450390
2. 671737369
0.8551833457
It is known that the behavior of the solution with the Maxwellian condition (17) depends on the values of. So we have considered various values for. The computations are carried out for the cases and 0.005 which are used in the earlier papers [15] [19] . The numerical conserved quantities with and 0.005 are given in Table 5. It is observed that the obtained values of the invariants remain almost constant during the computer run.
5. Conclusion
In this paper we study the MEW problem by extending the use of multigrid technique. We checked our scheme through single solitary wave in which the analytic solution is known. Our scheme was extended to study the interaction of two solitary waves and Maxwellian initial condition where the analytic solutions are unknown during the interaction. The performance and accuracy of the method were explained by calculating the error norms and conservative properties of mass, momentum and energy. The computed results showed that our scheme is a successful numerical technique for solving the MEW problem and can be also efficiently applied for solving a large number of physically important non-linear problems.
Cite this paper
Yasser M. Abo Essa, (2016) Multigrid Method for the Numerical Solution of the Modified Equal Width Wave Equation. Applied Mathematics,07,1140-1147. doi: 10.4236/am.2016.710102
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