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In this paper, we obtained the numerical solutions of the modified regularized long-wave (MRLW) equation , by using the multigrid method and finite difference method. The solitary wave motion, interaction of two and three 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 techniques.

The numerical solution of partial differential equations requires some discretization of the domain into a collection of points. A large system of equations comes out from discretization of the same partial differential equations and the optimal method for solving these problems is multigrid method, see [

Consider the following one-dimensional modified regularized long-wave (MRLW) equation: equation:

where

Although the analytical solutions of the MRLW equation, with a limited set of boundary and initial conditions, have been existed, many authors are recently interested in the numerical solutions of this equation. Gardner et al. [

An outline of this paper is as follows: we begin in Section 2 by reviewing the analytical solution of the MRLW 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 MRLW equation. Finally, in Section 4, we introduce the numerical results for solving the MRLW equation through some well known standard problems.

The exact solution of Equation (1) can be written in the form [

which represents the motion of a single solitary wave with amplitude

are arbitrary constants. The initial condition is given by

The conservation properties of the MRLW equation related to mass, momentum and energy are determined by following three invariants on the region

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

pressing in the form

where

Step 1:

Step 2: Starting from

The right hand side for the last equation 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

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 and three solitary waves and development of Maxwellian initial condition into solitary waves.

Consider equation (1) with boundary conditions

and the initial condition (4).

The analytical values of the invariants of this problem can be found as [

For a comparison with earlier studies [

Consider the interaction of two separated solitary waves having different amplitudes and travelling in the same direction as a second problem. For this problem, the initial condition is given by:

where

0 | 4.442882932 | 3.299703879 | 1.414341330 | 0.000000000 | 0.000000 |

1 | 4.442882973 | 3.299730717 | 1.414368263 | 2.971968997 | 1.685964 |

2 | 4.442882949 | 3.299730715 | 1.414368247 | 2.971975650 | 1.680891 |

3 | 4.442882955 | 3.299730689 | 1.414368233 | 2.971979150 | 1.687715 |

4 | 4.442882979 | 3.299730715 | 1.414368264 | 2.971987783 | 1.689784 |

5 | 4.442882963 | 3.299730703 | 1.414368245 | 2.971953900 | 1.686949 |

6 | 4.442882973 | 3.299730707 | 1.414368249 | 2.971968915 | 1.679297 |

7 | 4.442882961 | 3.299730688 | 1.414368229 | 2.971988844 | 1.686899 |

8 | 4.442882973 | 3.299730715 | 1.414368258 | 2.971957839 | 1.689770 |

9 | 4.442882975 | 3.299730701 | 1.414368235 | 2.971927885 | 1.687768 |

10 | 4.442882953 | 3.299730705 | 1.414368244 | 2.972006686 | 1.680656 |

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

Analytical | 4.4428829 | 3.2998316 | 1.4142135 | 0 | 0 |

Present | 4.4428829 | 3.2997307 | 1.4143682 | 0.297201 | 0.1680656 |

[ | 4.4431758 | 3.3003023 | 1.4146927 | 2.41552 | 1.07974 |

Cubic B-splines coll-CN [ | 4.442 | 3.299 | 1.413 | 16.39 | 9.24 |

Cubic B-splines coll+PA-CN [ | 4.440 | 3.296 | 1.411 | 20.3 | 11.2 |

Cubic B-splines coll [ | 4.44288 | 3.29983 | 1.41420 | 9.30196 | 5.43718 |

MQ [ | 4.4428829 | 3.29978 | 1.414163 | 3.914 | 2.019 |

IMQ [ | 4.4428611 | 3.29978 | 1.414163 | 3.914 | 2.019 |

IQ [ | 4.4428794 | 3.29978 | 1.414163 | 3.914 | 2.019 |

GA [ | 4.4428829 | 3.29978 | 1.414163 | 3.914 | 2.019 |

TPS [ | 4.4428821 | 3.29972 | 1.414104 | 4.428 | 2.306 |

For the computational discussion, we use parameters

In this section, the behavior of the interaction of three solitary waves having different amplitudes and travelling in the same direction was studied. So, we consider Equation (1) with the initial condition given by the linear sum of three well-separated solitary waves of different amplitudes:

where

For the computational work, we used parameters

Finally, the development of the Maxwellian initial condition:

into a train of solitary waves is discussed. It is known that the behavior of the solution with the Maxwellian condition (16) depends on the values of

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

T | ||||||

0 | 6.34543 | 0.592826 | 0.0054854 | 6.34543 | 0.592826 | 0.0054854 |

0.2 | 6.34540 | 0.592805 | 0.0054848 | 6.34541 | 0.592826 | 0.0054854 |

0.4 | 6.34541 | 0.592884 | 0.0054841 | 6.34541 | 0.592826 | 0.0054854 |

0.6 | 6.34541 | 0.592863 | 0.0054835 | 6.34541 | 0.592826 | 0.0054854 |

0.8 | 6.34541 | 0.592843 | 0.0054828 | 6.34542 | 0.592826 | 0.0054854 |

1.0 | 6.34541 | 0.592822 | 0.0054821 | 6.34542 | 0.592826 | 0.0054854 |

1.2 | 6.34542 | 0.592801 | 0.0054815 | 6.34542 | 0.592826 | 0.0054853 |

1.4 | 6.34542 | 0.592880 | 0.0054808 | 6.34542 | 0.592827 | 0.0054851 |

1.6 | 6.34542 | 0.592860 | 0.0054802 | 6.34541 | 0.592828 | 0.0054841 |

1.8 | 6.34542 | 0.592839 | 0.0054895 | 6.34540 | 0.592830 | 0.0054814 |

2.0 | 6.34542 | 0.592818 | 0.0054889 | 6.34540 | 0.592832 | 0.0054796 |

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

t | ||||||

0 | 9.51777 | 0.9041368 | 0.0078632 | 9.51777 | 0.9041368 | 0.0078632 |

0.1 | 9.51776 | 0.9041371 | 0.0078632 | 9.51766 | 0.9041370 | 0.0078630 |

0.2 | 9.51776 | 0.9041372 | 0.0078632 | 9.51766 | 0.9041370 | 0.0078631 |

0.3 | 9.51776 | 0.9041372 | 0.0078631 | 9.51767 | 0.9041369 | 0.0078631 |

0.4 | 9.51775 | 0.9041372 | 0.0078631 | 9.51767 | 0.9041369 | 0.0078631 |

0.5 | 9.51775 | 0.9041372 | 0.0078631 | 9.51768 | 0.9041369 | 0.0078631 |

0.6 | 9.51775 | 0.9041371 | 0.0078631 | 9.51768 | 0.9041369 | 0.0078632 |

0.7 | 9.51775 | 0.9041371 | 0.0078631 | 9.51768 | 0.9041368 | 0.0078632 |

0.8 | 9.51774 | 0.9041371 | 0.0078630 | 9.51768 | 0.9041369 | 0.0078632 |

0.9 | 9.51774 | 0.9041371 | 0.0078630 | 9.51768 | 0.9041372 | 0.0078628 |

1.0 | 9.51774 | 0.9041373 | 0.0078630 | 9.51768 | 0.9041384 | 0.0078616 |

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

t | |||||||

0.015 | 0.01 | 1.77247 | 1.27212 | 0.867431 | 1.77247 | 1.27212 | 0.867430 |

0.03 | 1.77247 | 1.27210 | 0.867429 | 1.77246 | 1.27207 | 0.867341 | |

0.05 | 1.77247 | 1.27209 | 0.867427 | 1.77243 | 1.27296 | 0.867156 | |

0.004 | 0.01 | 1.77247 | 1.25833 | 0.881213 | 1.77247 | 1.25833 | 0.881212 |

0.03 | 1.77247 | 1.25831 | 0.881210 | 1.77246 | 1.25827 | 0.881091 | |

0.05 | 1.77247 | 1.25828 | 0.881209 | 1.77246 | 1.25819 | 0.880750 |

In this work we extended the use of multigrid technique to initial boundary value problems, namely the MRLW problem. We tested our scheme through single solitary wave in which the analytic solution is known. Our scheme was extended to study the interaction of two and three solitary waves and Maxwellian initial condition where the analytic solutions are unknown during the interaction. The performance and accuracy of the method were shown by calculating the error norms