_{1}

^{*}

In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The
L
_{2} and
L
_{∞} error norms are computed to study the accuracy and the simplicity of the presented method.

Discretization using finite differences in time and spectral methods in space has proved to be very useful in solving numerically non-linear Partial Differential Equations (PDEs) describing wave propagation. The Korteweg de Vries (KdV) equation is one famous example to which such combined schemes have been applied efficiently to analyze efficiently unidirectional solitary wave propagation in one dimension [

where μ is a positive parameter and the subscripts x and t denote differentiation, when solved analytically, within an infinite region with physical boundary conditions

and the initial condition

where

For the numerical treatment, the spatial variable x of Equation (1) is restricted over an interval

where

For ease of presentation the spatial period [a, b] is normalized to [0, 2π], with the change of variable

Let L = b − a. Thus, Equations (4) and (5) become

In practice, we need to discretize Equations (6) and (7). For any integer N > 0, consider

From this, using the inversion formula, we get

Replacing F and

Letting

where G(U) defines the right hand side of Equation (13).

A time integration known as a Leap-Frog method (a two-step scheme) is given as

Use the Leap-Frog scheme to advance in time, we obtain

This is called the Fourier-Leap-Frog (FLF) scheme for the MEW Equation (14). FLF method needs two levels of initial data, we begin with

Then evaluate second level of initial solution

then substitute

to get

We substitute V(x, 0) and U(x, Δt) in Equation (19) to evaluate V(x, 2Δt) then substitute V(x, 2Δt) in Equation (18) to evaluate U(x, 2Δt), so we have V(x, Δt) and U(x, 2Δt), substitute in Equation (19) to evaluate V(x, 3Δt) and evaluate U(x, 3Δt) from Equation (18) and so on, until we evaluate U(x, t) at time t = nΔt.

In order to show how good the numerical solutions are in comparison with the exact ones, L_{2} and L_{∞} error norms will be computed by

The conservation properties of the MEW equation will be examined by calculating the following three invariants, given as [

For the computation of U_{x} in Equation (21), we used Fourier transform. To implement the performance of the method, three test problems will be considered: the motion of a single solitary wave, development of two positive solitary waves interaction, development of three positive solitary wave interaction.

Consider Equation (1) with the boundary

This problem has a solitary wave solution of the form

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

For the numerical simulation of the motion of a single solitary wave, the parameters Δx = 0.1, Δt = 0.001, μ = 1, x_{0} = 30, N = 2048 and A = 0.25 are chosen. The analytical values for the invariants are C_{1} = 0.7853982, C_{2} = 0.1666667, and C_{3} = 0.0052083. As it is seen from _{1} and C_{3} remain almost constant during the computer run at times t = 0 to t = 100 (changes of the invariants C_{1} and C_{3} approach zero), where C_{2} changes from its initial value by less than 1 × 10^{−}^{9}. The error norms L_{2} and L_{∞} at different various times are shown in ^{−}^{5}. Error distribution at time t = 20 is drawn in _{0} = 30 and A = 0.25. As it is seen from _{1}, C_{2} and C_{3} closed to the analytical values when N increases. The comparison between the results obtained by the present with those in the other studies [

t | C_{1} | C_{2} | C_{3} | L_{2} × 10^{3} | L_{∞} × 10^{3} |
---|---|---|---|---|---|

0 | 0.785014668 | 0.166625987 | 0.005205790 | 0.0000000 | 0.0000000 |

5 | 0.785014668 | 0.166625987 | 0.005205790 | 0.0069317 | 0.0032278 |

10 | 0.785014668 | 0.166625986 | 0.005205790 | 0.0160855 | 0.0081248 |

15 | 0.785014668 | 0.166625986 | 0.005205790 | 0.0226029 | 0.0113299 |

20 | 0.785014668 | 0.166625986 | 0.005205790 | 0.0275859 | 0.0129414 |

25 | 0.785014668 | 0.166625986 | 0.005205790 | 0.0343768 | 0.0161599 |

30 | 0.785014668 | 0.166625986 | 0.005205790 | 0.0410980 | 0.0193344 |

35 | 0.785014668 | 0.166625986 | 0.005205790 | 0.0477366 | 0.0224764 |

40 | 0.785014668 | 0.166625986 | 0.005205790 | 0.0558158 | 0.0273561 |

45 | 0.785014668 | 0.166625986 | 0.005205790 | 0.0622846 | 0.0305792 |

50 | 0.785014668 | 0.166625986 | 0.005205790 | 0.0686557 | 0.0337939 |

100 | 0.785014668 | 0.166625986 | 0.005205790 | 0.1239589 | 0.0654669 |

N | C_{1} | C_{2} | C_{3} | L_{2} × 10^{3} | L_{∞} × 10^{3} |
---|---|---|---|---|---|

512 | 0.7838642 | 0.1665041 | 0.0051982 | 0.1103535 | 0.0517942 |

1024 | 0.7846312 | 0.1665853 | 0.0052033 | 0.0551734 | 0.0258918 |

2048 | 0.7850147 | 0.1666260 | 0.0052058 | 0.0275859 | 0.0129412 |

4096 | 0.7852064 | 0.1666463 | 0.0052071 | 0.0137927 | 0.0064692 |

8192 | 0.7853023 | 0.1666565 | 0.0052077 | 0.0068963 | 0.0032342 |

Ref [ | 0.7853898 | 0.1667614 | 0.0052082 | 0.0796940 | 0.0465523 |

Ref [ | 0.7849545 | 0.1664765 | 0.0051995 | 0.2905166 | 0.2498925 |

Ref [ | 0.7853977 | 0.1664735 | 0.0052083 | 0.2692812 | 0.2569972 |

Ref [ | 0.7853967 | 0.1666663 | 0.0052083 | 0.0800980 | 0.0460618 |

The initial condition given by the linear sum of two separate solitary waves of various amplitudes

where_{1} = 1, A_{2} = 0.5, x_{1} = 15, x_{2} = 30, N = 8192, Δx = 0.1 and Δt = 0.01. The analytic invariants are [

position x = 34.7 and of the larger wave 1.000097 having the position x = 44.4 are measured at time t = 55 so that difference in amplitudes is 0.010741 for the smaller wave and 0.000097 for the larger wave. _{2} is 6.11 × 10^{−}^{5} and in C_{3} is 5.68 × 10^{−}^{5} and C_{1} is exact up to the last recorded digit.

The intersection of two solitary waves was also studies with the following parameters: μ = 1, x_{1} = 15, x_{2} = 30, A_{1} = −2, A_{2} = 1, N = 8192, Δt = 0.01 and Δx = 0.1 in the range 0 ≤ x ≤ 819.2. The experiment was run from t = 0 to t = 55 to allow the interaction to take place. _{1} = −3.1415927, C_{2} = 13.3333333 and C_{2} = 22.6666667. It can be seen in

Interaction of three solitary waves is studied by considering Equation (1) with the following initial condition:

where_{1} = 1, A_{2} = 0.5, A_{3} = 0.25, x_{1} = 15, x_{2} = 30, x_{3} = 45, N = 8192, Δx = 0.1 and Δt = 0.01. Solitary wave having the largest amplitude is located to the left of the smaller ones. As is well known, solitary waves with larger amplitudes have a greater velocity than those with smaller amplitudes. Consequently, as time goes on the larger two solitary waves catches up with the smaller one, the overlapping process of the three solitary waves continues while the larger solitary waves have overtaken the smaller ones. Plot of the three solitary waves is depicted at various times in

t | A_{1} = 1, A_{2} = 0.5 | A_{1} = −2, A_{2} = 1 | ||||
---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{1} | C_{2} | C_{3} | |

0 | 4.7118132 | 3.3331323 | 1.4164968 | −3.1412080 | 13.3325098 | 22.6638508 |

5 | 4.7118132 | 3.3330876 | 1.4164524 | −3.1412080 | 13.3293775 | 22.6514303 |

10 | 4.7118132 | 3.3330830 | 1.4164479 | −3.1412080 | 13.3306037 | 22.6526691 |

15 | 4.7118132 | 3.3330828 | 1.4164480 | −3.1412080 | 13.3293817 | 22.6514527 |

20 | 4.7118132 | 3.3330828 | 1.4164480 | −3.1412080 | 13.3293781 | 22.6514501 |

25 | 4.7118132 | 3.3330822 | 1.4164476 | −3.1412080 | 13.3293776 | 22.6514500 |

30 | 4.7118132 | 3.3330776 | 1.4164476 | −3.1412080 | 13.3293774 | 22.6514500 |

35 | 4.7118132 | 3.3330712 | 1.4164400 | −3.1412080 | 13.3293774 | 22.6514500 |

40 | 4.7118132 | 3.3330803 | 1.4164462 | −3.1412080 | 13.3293773 | 22.6514500 |

45 | 4.7118132 | 3.3330826 | 1.4164478 | −3.1412080 | 13.3293749 | 22.6514500 |

50 | 4.7118132 | 3.3330831 | 1.4164481 | −3.1412080 | 13.3293750 | 22.6514500 |

55 | 4.7118132 | 3.3330831 | 1.4164481 | −3.1412080 | 13.3293751 | 22.6514500 |

waves can be openly observed from the time-amplitude graph in _{2} is 5.37 × 10^{−}^{5} and in C_{3} is 5.09 × 10^{−}^{5} and C_{1} is exact up to the last recorded digit. The analytical values can be found [

We consider here is the numerical solution of the Equation (1) with the Maxwellian initial condition

with the boundary conditions

As it is known, Maxwellian initial condition the behavior of the solution depends on the values of µ. The computations are carried out for the cases µ = 1, 0.5, 0.1, 0.05, 0.02 and 0.005 which are used in [

The Fourier Leap Frog method has been successfully applied to obtain the numerical solution of the modified equal width wave equation. Four test problems are worked out to examine the performance of the used method. The motion of a single solitary wave and its accuracy was shown by calculating error norms L_{2} and L_{∞} and shown in the figures and tables. The interaction of two solitary waves and its accuracy shown by compare with other numerical solutions. The interaction of three solitary waves and its accuracy shown by compare with other numerical solutions. A Maxwellian initial condition pulse is then studied at different values of µ. The invariants

t | C_{1} | C_{2} | C_{3} |
---|---|---|---|

0 | 5.4971155 | 3.4997894 | 1.4217047 |

20 | 5.4971155 | 3.4997399 | 1.4216558 |

40 | 5.4971155 | 3.4997374 | 1.4216540 |

60 | 5.4971155 | 3.4997357 | 1.4216538 |

80 | 5.4971155 | 3.4997401 | 1.4216560 |

100 | 5.4971155 | 3.4997401 | 1.4216560 |

120 | 5.4971155 | 3.4997401 | 1.4216560 |

140 | 5.4971155 | 3.4997400 | 1.4216559 |

160 | 5.4971155 | 3.4997401 | 1.4216560 |

180 | 5.4971155 | 3.4997401 | 1.4216560 |

200 | 5.4971155 | 3.4997401 | 1.4216560 |

t | µ | C_{1} | C_{2} | C_{3} |
---|---|---|---|---|

0 | 1 | 1.7715884 | 2.5066286 | 0.8857942 |

3 | 1.7715884 | 2.5066279 | 0.8857938 | |

6 | 1.7715884 | 2.5066276 | 0.8857936 | |

9 | 1.7715884 | 2.5066277 | 0.8857937 | |

12 | 1.7715884 | 2.5066275 | 0.8857937 | |

0 | 0.5 | 1.7715884 | 1.8796654 | 0.8857942 |

3 | 1.7715884 | 1.8796645 | 0.8857934 | |

6 | 1.7715884 | 1.8796646 | 0.8857934 | |

9 | 1.7715884 | 1.8796646 | 0.8857935 | |

12 | 1.7715884 | 1.8796645 | 0.8857935 | |

0 | 0.1 | 1.7715884 | 1.3780948 | 0.8857942 |

3 | 1.7715884 | 1.3780956 | 0.8857942 | |

6 | 1.7715884 | 1.3780957 | 0.8857942 | |

9 | 1.7715884 | 1.3780955 | 0.8857941 | |

12 | 1.7715884 | 1.3780955 | 0.8857941 | |

0 | 0.05 | 1.7715884 | 1.3153985 | 0.8857942 |

3 | 1.7715884 | 1.3154016 | 0.8857967 | |

6 | 1.7715884 | 1.3154017 | 0.8857967 | |

9 | 1.7715884 | 1.3154017 | 0.8857967 | |

12 | 1.7715884 | 1.3154016 | 0.8857967 | |

0 | 0.02 | 1.7715884 | 1.2777807 | 0.8857942 |

3 | 1.7715884 | 1.2777913 | 0.8858065 | |

6 | 1.7715884 | 1.2777914 | 0.8858066 | |

9 | 1.7715884 | 1.2777913 | 0.8858066 | |

12 | 1.7715884 | 1.2777914 | 0.8858066 | |

0 | 0.005 | 1.7715884 | 1.2589718 | 0.8857942 |

3 | 1.7715884 | 1.2590204 | 0.8858617 | |

6 | 1.7715884 | 1.2590208 | 0.8858619 | |

9 | 1.7715884 | 1.2590209 | 0.8858619 | |

12 | 1.7715884 | 1.2590209 | 0.8858619 |

are satisfactorily constant in computer run in all cases. The obtained results show that the present method is a remarkably successful numerical method and can also be efficiently applied to other types of non-linear problems.

Hany N. Hassan, (2016) An Accurate Numerical Solution for the Modified Equal Width Wave Equation Using the Fourier Pseudo-Spectral Method. Journal of Applied Mathematics and Physics,04,1054-1067. doi: 10.4236/jamp.2016.46110