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 [1] -[9] .

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

where is the time, is the space coordinate, are positive constants and is the wave amplitude with the physical boundary conditions as. This equation was first introduced to describe the development of an undular bore by Peregrine [10] and later by Benjamin et al. [11] . Equation (1) has various applications as in physics media since it describes the phenomena with weak nonlinearity and dispersion waves, including nonlinear transverse waves in shallow water, ion-acoustic and magneto hydrodynamic waves in plasma, and phonon packets in nonlinear crystals [11] .

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. [12] introduced a collocation solution to the MRLW equation using quintic B-spline finite elements. Khalifa et al. [13] [14] applied the finite difference and cubic B-spline collocation finite element method to obtain the numerical solutions of the MRLW equation. Solutions based on collocation method with quadratic B-spline finite elements and the central finite difference method for time are investigated by Raslan [15] . Raslan and Hassan [16] solved the MRLW equation by a collocation finite element method using quadratic, cubic, quartic, and quintic B-spline to obtain the numerical solutions of the single solitary wave. Ali [17] has formulated a classical radial basis function collocation method for solving the MRLW equation. Haq et al. [18] have developed a numerical scheme based on quartic B-spline collocation method for the numerical solution of MRLW equation. Karakoc and Geyikli [19] solved the MRLW equation by using the Petrov-Galerkin finite element method.

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 [12] [14] :

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

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] . 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. 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 ex-

pressing 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 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, 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 on the fine grid with.

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

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. and error norms are used to show how good the numerical results in comparison with the exact results.

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, and conservative properties of mass, momentum and energy. The computed results showed that the present scheme is a successful numerical technique for solving the MRLW problem.