_{1}

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 [

The modified equal width wave (MEW) equation introduced by Morrison et al. [

Several solutions for MEW had been proposed in [

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.

The modified equal width wave equation which is as a model for non-linear dispersive waves, considered here has the normalized form [

with the physical boundary conditions

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 [

which represents the motion of a single solitary wave with amplitude A, where the wave velocity

For the MEW equation, it is important to discuss the following three invariant conditions given in [

The basic idea of multigrid techniques is illustrated by Brandt [_{1} and use these solutions as initial values for the next level

We apply the full multigrid algorithm for the MRLW equation. Assuming the initial condition

We start handling the non-linear term

space difference for Equation (1) is

where

Step 1:

Step 2: Starting from

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

that can be written explicitly as:

Step 4: Doing relaxation sweep on

Step 5: Computing the residuals

Step 6: Computing an approximate solution of error

Step 7: Interpolating the solution of error

By taking this solution on coarse grid and repeating steps 3-7, we obtain the approximate values of u on the grid with

Step 8:

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 [

The accuracy of the method is measured by both the

and the

to show how good the numerical results in comparison with the exact results.

Consider Equation (1) with boundary conditions (2) and the initial condition (4). For a comparison with earlier studies [

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

In [

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 |

Method | |||||
---|---|---|---|---|---|

Analytical | 0.7853982 | 0.1666667 | 0.0052083 | 0 | 0 |

Present | 0.7853966 | 0.1666638 | 0.0052082 | 0.520804 | 0.54566 |

[ | 0.7853898 | 0.1667614 | 0.0052082 | 7.969400 | 4.65523 |

[ | 0.7849545 | 0.1664765 | 0.0051995 | 29.05166 | 24.98925 |

[ | 0.7853977 | 0.1664735 | 0.0052083 | 26.92812 | 25.69972 |

[ | 0.7853967 | 0.1666663 | 0.0052083 | 8.009800 | 4.606180 |

Present method | [ | |||||
---|---|---|---|---|---|---|

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 |

[

Finally, we have studied the interaction of two solitary waves with the following parameters

The analytical invariants can be found as in [

Last study, we consider the numerical solution of the equation (1) with the Maxwellian initial condition

and the boundary conditions

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 |

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

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

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