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Numerical solution of the modified equal width wave equation is obtained by using lumped Galerkin method based on cubic B-spline finite element method. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. Accuracy of the proposed method is discussed by computing the numerical conserved laws

The modified equal width wave equation (MEW) based upon the equal width wave (EW) equation [

The modified equal width wave (MEW) equation considered here has the normalized form [

The stability analysis is based on the Von Neumann theory in which the growth factor

In this part, we consider the following two test problems: the motion of a single solitary wave and interaction of two solitary waves. All computations are executed on a Pentium 4 PC in the Fortran code using double precision arithmetic. For the MEW equation, it is important to discuss the following three invariant conditions given in [

For this problem we consider (

Invariants and error norms for single solitary wave with

0 | 0.7853966 | 0.1666661 | 0.0052083 | 0.0000000 | 0.0000000 |

5 | 0.7853966 | 0.1666662 | 0.0052083 | 0.0204838 | 0.0115451 |

10 | 0.7853966 | 0.1666662 | 0.0052083 | 0.0407743 | 0.0231561 |

15 | 0.7853967 | 0.1666662 | 0.0052083 | 0.0606975 | 0.0347169 |

20 | 0.7853967 | 0.1666663 | 0.0052083 | 0.0800980 | 0.0460618 |

20 [ | 0.7853898 | 0.1667614 | 0.0052082 | 0.0796940 | 0.0465523 |

20 [ | 0.7849545 | 0.1664765 | 0.0051995 | 0.2905166 | 0.2498925 |

20 [ | 0.7853977 | 0.1664735 | 0.0052083 | 0.2692812 | 0.2569972 |

The order of convergence at

order | order | |||
---|---|---|---|---|

0.8 | 4.16296467 | — | 2.78665608 | — |

0.4 | 1.21228931 | 1.77987727 | 0.68462204 | 2.02515531 |

0.2 | 0.31752640 | 1.93278558 | 0.18175645 | 1.91330117 |

0.1 | 0.08009801 | 1.9803824 | 0.04606181 | 1.98036355 |

0.05 | 0.01932448 | 2.05133680 | 0.01122974 | 2.03624657 |

0.025 | 0.00530959 | 1.86375722 | 0.00304044 | 1.88497250 |

The order of convergence at

order | order | |||
---|---|---|---|---|

0.8 | 0.08383192 | — | 0.08300304 | — |

0.4 | 0.07424489 | 0.17520793 | 0.05061152 | 0.71369837 |

0.2 | 0.07833790 | −0.07817543 | 0.04448503 | 0.18614587 |

0.1 | 0.07972878 | −0.02539013 | 0.04573242 | −0.03989733 |

0.05 | 0.08009801 | −0.00666580 | 0.04606181 | −0.01035383 |

0.025 | 0.08019167 | −0.00168598 | 0.04614408 | −0.00257446 |

The motion of a single solitary wave with

In this section, we consider (

Firstly we studied the interaction of two positive solitary waves with the parameters

Invariants for the interaction of two solitary waves.

0 | 4.7123732 | 3.3333253 | 1.4166643 | −3.1415737 | 13.3332816 | 22.6665313 |

5 | 4.7123861 | 3.3333482 | 1.4166852 | −3.1458603 | 13.3449843 | 22.7133525 |

10 | 4.7123959 | 3.3333621 | 1.4166982 | −3.1377543 | 13.3031153 | 22.5832812 |

15 | 4.7124065 | 3.3333785 | 1.4167141 | −3.1625436 | 13.3838991 | 22.8926223 |

20 | 4.7124249 | 3.3334164 | 1.4167521 | −3.1658318 | 13.3999304 | 22.9451481 |

25 | 4.7124899 | 3.3335832 | 1.4169238 | −3.1701819 | 13.4126391 | 22.9954601 |

30 | 4.7127643 | 3.3333557 | 1.4177617 | −3.1747553 | 13.4251358 | 22.0458834 |

35 | 4.7130474 | 3.3352500 | 1.4188849 | −3.1793707 | 13.4376785 | 23.0966349 |

40 | 4.7124881 | 3.3336316 | 1.4171690 | −3.1840126 | 13.4502895 | 23.1477374 |

45 | 4.7123002 | 3.3331878 | 1.4167580 | −3.1886789 | 13.4629730 | 23.1991979 |

50 | 4.7122479 | 3.3330923 | 1.4167142 | −3.1933699 | 13.4757303 | 23.2510209 |

55 | 4.7122576 | 3.3331149 | 1.4167237 | −3.1980856 | 13.4885624 | 23.3032108 |

Comparison of invariants for the interaction of two solitary waves with results from [

0 | 4.7123732 | 3.3333253 | 1.4166643 | 4.7123884 | 3.3352890 | 1.4166697 |

5 | 4.7123861 | 3.3333482 | 1.4166852 | 4.7123718 | 3.3352635 | 1.4166486 |

10 | 4.7123959 | 3.3333621 | 1.4166982 | 4.7123853 | 3.3352836 | 1.4166647 |

15 | 4.7124065 | 3.3333785 | 1.4167141 | 4.7123756 | 3.3352894 | 1.4166772 |

20 | 4.7124249 | 3.3334164 | 1.4167521 | 4.7123748 | 3.3353041 | 1.4166926 |

25 | 4.7124899 | 3.3335832 | 1.4169238 | 4.7124173 | 3.3354278 | 1.4168363 |

30 | 4.7127643 | 3.3333557 | 1.4177617 | 4.7126410 | 3.3359464 | 1.4176398 |

35 | 4.7130474 | 3.3352500 | 1.4188849 | 4.7128353 | 3.3364247 | 1.4186746 |

40 | 4.7124881 | 3.3336316 | 1.4171690 | 4.7123946 | 3.3355951 | 1.4170695 |

45 | 4.7123002 | 3.3331878 | 1.4167580 | 4.7122273 | 3.3352364 | 1.4166637 |

50 | 4.7122479 | 3.3330923 | 1.4167142 | 4.7121567 | 3.3351175 | 1.4165797 |

55 | 4.7122576 | 3.3331149 | 1.4167237 | 4.7121400 | 3.3350847 | 1.4165527 |

Interaction of two solitary waves at different times.

Interaction of two solitary waves at different times.

In this paper, the cubic B-spline lumped Galerkin method has been successfully applied to obtain the numerical solution of the modified equal width wave equation. The efficiency of the method was tested on two test problems of wave propagation: the motion of a single solitary wave and the interaction of two solitary waves, and its accuracy was shown by calculating error norms