^{1}

^{*}

^{2}

^{1}

^{3}

The improved Boussinesq equation is solved with classical finite element method using the most basic Lagrange element k = 1, which leads us to a second order nonlinear ordinary differential equations system in time; this can be solved by any standard accurate numerical method for example Runge-Kutta-Fehlberg. The technique is validated with a typical example and a fourth order convergence in space is confirmed; the 1- and 2-soliton solutions are used to simulate wave travel, wave splitting and interaction; solution blow up is described graphically. The computer symbolic system MathLab is quite used for numerical simulation in this paper; the known results in the bibliography are confirmed.

The improved Boussinesq equation (IBq) was proposed in Bogolyubsky’s work [

where

which is the IBq and will be the principal study equation of this paper; it is convenient for computer simulation of the dynamics of different nonlinear waves with weak dispersion; in our case the IBq equation will help to formulate the finite element discretization in the spatial direction with the primal L_{2}-Galerkin finite element formulation [

where

where

The boundary conditions at

Linearization techniques and finite differences are employed in most numerical works that solve the IBq [

contrast with the method proposed in this paper such a restriction is not needed. The nonlinear term

which for the finite difference method is a problem and needs to be linearized with the help of bounds solutions and/or iterative approach [_{2}-Galerkin finite element formulation and leads us due to the reduced support in the basis functions to a time dependent tridia- gonal antisymmetric square matrix for the

better convergence properties in the x direction. The

head-on collision, and blow-up solution are modeled and graphics representations are done [

The classical finite element method relies over two basic ingredients [

where

the subindex and superindex 0, 1 refers to boundary conditions and to the derivative order that should belong to

Find

A classical (or conforming) approximation of u is obtained by looking for a function

The second basic ingredient for the classical finite element method is to choose the finite dimensional subspace

this end let

is continuos on

In this way for each

_{2}-Galerkin space semi-discretization for the IBq equation. Find

If we substitute

where

As is usual all finite element computations like integration, interpolation are done over the master element

they have the property

The local finite element matrices are calculated over

by the respective scale factor, to get the finite element matrix over

typical finite element assembly to get the global matrices

transforms to

and after assembly from element 2 to N, M is given as follows

over

analogously for K whose scale factor for integration is

the matrix

finally after assembly and putting the boundary conditions

this matrix represents the nonlinearity in the IBq Equation (2), the anti-symmetry structure is related to the

With the matrices

the matrix

the next first order nonlinear system of ordinary differential equations:

with initial conditions

the system (14) and (15), (16) is a standard initial value problem that can now be solved by integration algorithms like predictor corrector [

Firstly in Numerical Validation, the proposed method is used for the numerical wave propagation simulation, and comparing this simulation with the exact solution we validate the method, we are really approximating the soliton solution by a non-classical one, the compacton [

1. Numerical Validation. We set

the exact solution is given by (3), we discretize over

are compared for t = 20 with the exact solution at some points in

2. Wave brake-up. With the same

x | Error | ||
---|---|---|---|

15.1124 | 0.0358 | 0.0356 | 0.0002 |

20.0583 | 0.2944 | 0.2949 | 0.0005 |

22.9059 | 0.4990 | 0.4988 | 0.0002 |

25.0042 | 0.4016 | 0.4013 | 0.0003 |

30.0999 | 0.0567 | 0.0567 | 0.0000 |

35.0458 | 0.0050 | 0.0050 | 0.0000 |

3. The head-on wave collision. In this example we take

A negative speed indicate a wave traveling to the negative x side direction, so the two waves will have a head-on collision [

The next examples are done with different amplitudes

If

If

If

If

These results are in good agreement with those reported elsewhere [

4. Blow-up solution. The blow-up solution is now simulated as discussed in [

numerically on

It is know [

5. Convergence Order. For our technique, the convergence order will be calculated in the usual way using the results from Numerical Validation, as the following

A concrete development of a practical

Number of elements | Error in | C.O. |
---|---|---|

20 | 0.38524837209495 | |

40 | 0.05864516158548 | 2.715704864 |

80 | 0.00361236523776 | 4.020996413 |

brake-up result if the initial pulse is steady. The head-on collision is successfully simulated to different wave amplitudes to obtain the existence of a critical value 0.5. If the amplitudes are below or even equal to this critical value, the head-on collision is elastic and the graphics show a clean interaction before and after the collision. If one or two of the amplitudes are greater than the critical value, the head-on collision is inelastic and the graphics show a secondary soliton interaction. It has been verified numericaly the existence of a blow-up solution in finite time to a theoretical problem and was noted that for the

dependent matrix called

This work was supported in part by the Secretary of Education of Mexico under the project PROMEP 103.5/13/9347 for developing research scientific groups. This effort is greatly appreciated.