_{1}

In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including; single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms,
*L*
_{2} and
*L*
_{∞} and invariants
*I*
_{1},
*I*
_{2} and
*I*
_{3} have been calculated. Our numerical results are compared with some of those available in the literature.

Several physical processes for example dispersion of long waves in shallow water waves under gravity, bubble-liquid mixtures, ion acoustic plasma waves, fluid mechanics, nonlinear optics and wave phenomena in enharmonic crystals can be expressed by the KdV equation which was first introduced by Korteweg and de Vries [

In this article, we will take in consideration for the following GKdV equation

U t + ε U 2 U x + μ U x x x = 0 , (1)

with the homogeneous boundary conditions

U ( a , t ) = 0 , U ( b , t ) = 0 , U x ( a , t ) = 0 , U x ( b , t ) = 0 , t > 0 (2)

and an initial condition

U ( x , 0 ) = U 0 ( x ) , a ≤ x ≤ b , (3)

where t is time, x is the space coordinate, ε and μ are positive parameters. One of the primary mathematical models for describing the theory of water waves in shallow channels is the following Korteweg de-Vries (KdV) equation:

U t + ε U U x + μ U x x x = 0. (4)

The terms U U x and U x x x in the Equation (4) stand for the nonlinear convection and dispersion, respectively. In this paper, we have numerically solved the GKdV equation using collocation method with septic B-spline finite elements. We have investigated the motion of a single soliton wave to show the performance and profiency of the proposed method. Also we have showed the suggested method is unconditionally stable applying the von-Neumann stability analysis.

We think of a mesh a = x 0 < x 1 < ⋯ < x N = b as a uniform divide of the solution area a ≤ x ≤ b by the points x m with h = b − a N = x m + 1 − x m . The septic B-splines ϕ m ( x ) , ( m = − 3, − 2, ⋯ , N + 3 ) at the knots x m are given by [

ϕ m ( x ) = 1 h 7 { ( x − x m − 4 ) 7 [ x m − 4 , x m − 3 ] ( x − x m − 4 ) 7 − 8 ( x − x m − 3 ) 7 [ x m − 3 , x m − 2 ] ( x − x m − 4 ) 7 − 8 ( x − x m − 3 ) 7 + 28 ( x − x m − 2 ) 7 [ x m − 2 , x m − 1 ] ( x − x m − 4 ) 7 − 8 ( x − x m − 3 ) 7 + 28 ( x − x m − 2 ) 7 − 56 ( x − x m − 1 ) 7 [ x m − 1 , x m ] ( x m + 4 − x ) 7 − 8 ( x m + 3 − x ) 7 + 28 ( x m + 2 − x ) 7 − 56 ( x m + 1 − x ) 7 [ x m , x m + 1 ] ( x m + 4 − x ) 7 − 8 ( x m + 3 − x ) 7 + 28 ( x m + 2 − x ) 7 [ x m + 1 , x m + 2 ] ( x m + 4 − x ) 7 − 8 ( x m + 3 − x ) 7 [ x m + 2 , x m + 3 ] ( x m + 4 − x ) 7 [ x m + 3 , x m + 4 ] 0 otherwise (5)

The set of septic B-spline functions { ϕ − 3 ( x ) , ϕ − 2 ( x ) , ⋯ , ϕ N + 2 ( x ) , ϕ N + 3 ( x ) } forms a basis for the problem region of solution [ a , b ] . The approximate solution U N ( x , t ) to the exact solution U ( x , t ) in the form:

U N ( x , t ) = ∑ m = − 3 N + 3 ϕ m ( x ) δ m ( t ) (6)

where ϕ m ( x ) are septic B-splines and δ m ( t ) are time dependent parameters to be identified from the boundary and collocation conditions. A characteristic finite interval [ x m , x m + 1 ] is turn into the interval [ 0,1 ] by a domestic coordinate conversion described by h ξ = x − x m , 0 ≤ ξ ≤ 1 . So septic B-splines (5) in terms of ξ over [ 0,1 ] can be written as

ϕ m − 3 = 1 − 7 ξ + 21 ξ 2 − 35 ξ 3 + 35 ξ 4 − 21 ξ 5 + 7 ξ 6 − ξ 7 , ϕ m − 2 = 120 − 392 ξ + 504 ξ 2 − 280 ξ 3 + 84 ξ 5 − 42 ξ 6 + 7 ξ 7 , ϕ m − 1 = 1191 − 1715 ξ + 315 ξ 2 + 665 ξ 3 − 315 ξ 4 − 105 ξ 5 + 105 ξ 6 − 21 ξ 7 , ϕ m = 2416 − 1680 ξ + 560 ξ 4 − 140 ξ 6 + 35 ξ 7 ,

ϕ m + 1 = 1191 + 1715 ξ + 315 ξ 2 − 665 ξ 3 − 315 ξ 4 + 105 ξ 5 + 105 ξ 6 − 35 ξ 7 , ϕ m + 2 = 120 + 392 ξ + 504 ξ 2 + 280 ξ 3 − 84 ξ 5 − 42 ξ 6 + 21 ξ 7 , ϕ m + 3 = 1 + 7 ξ + 21 ξ 2 + 35 ξ 3 + 35 ξ 4 + 21 ξ 5 + 7 ξ 6 − ξ 7 , ϕ m + 4 = ξ 7 . (7)

Using Equation (5) and Equation (6), the nodal values of U m , U ′ m , U ″ m , U ‴ m and U m i v at the knots x m are obtained as the following:

U N ( x m , t ) = U m = δ m − 3 + 120 δ m − 2 + 1191 δ m − 1 + 2416 δ m + 1191 δ m + 1 + 120 δ m + 2 + δ m + 3 , U ′ m = 7 h ( − δ m − 3 − 56 δ m − 2 − 245 δ m − 1 + 245 δ m + 1 + 56 δ m + 2 + δ m + 3 ) , U ″ m = 42 h 2 ( δ m − 3 + 24 δ m − 2 + 15 δ m − 1 − 80 δ m + 15 δ m + 1 + 24 δ m + 2 + δ m + 3 ) , U ‴ m = 210 h 3 ( − δ m − 3 − 8 δ m − 2 + 19 δ m − 1 − 19 δ m + 1 + 8 δ m + 2 + δ m + 3 ) , U m i v = 840 h 4 ( δ m − 3 − 9 δ m − 1 + 16 δ m − 9 δ m + 1 + δ m + 3 ) (8)

where the symbols ' , ' ' , ' ' ' symbolize differentiation according to x, respectively. Using (5) and (8) in the Equation (1) this guides to a set of ordinary differential equations of the form

δ ˙ m − 3 + 120 δ ˙ m − 2 + 1191 δ ˙ m − 1 + 2416 δ ˙ m + 1191 δ ˙ m + 1 + 120 δ ˙ m + 2 + δ ˙ m + 3 + 7 ε Z m h ( − δ m − 3 − 56 δ m − 2 − 245 δ m − 1 + 245 δ m + 1 + 56 δ m + 2 + δ m + 3 ) + 210 μ h 3 ( − δ m − 3 − 8 δ m − 2 + 19 δ m − 1 − 19 δ m + 1 + 8 δ m + 2 + δ m + 3 ) = 0 , (9)

where

Z m = ( δ m − 3 + 120 δ m − 2 + 1191 δ m − 1 + 2416 δ m + 1191 δ m + 1 + 120 δ m + 2 + δ m + 3 ) 2 .

If time parameters δ i and its time derivatives δ ˙ i in Equation (9) are separated by the Crank-Nicolson form and finite difference approach, respectively:

δ i = δ i n + 1 + δ i n 2 , (10)

and usual finite difference approximation

δ ˙ i = δ i n + 1 − δ i n Δ t (11)

we acquired a repetition relationship between two time levels n and n + 1 relating two unknown parameters δ i n + 1 , δ i n for i = m − 3 , m − 2 , ⋯ , m + 2 , m + 3

γ 1 δ m − 3 n + 1 + γ 2 δ m − 2 n + 1 + γ 3 δ m − 1 n + 1 + γ 4 δ m n + 1 + γ 5 δ m + 1 n + 1 + γ 6 δ m + 2 n + 1 + γ 7 δ m + 3 n + 1 = γ 7 δ m − 3 n + γ 6 δ m − 2 n + γ 5 δ m − 1 n + γ 4 δ m n + γ 3 δ m + 1 n + γ 2 δ m + 2 n + γ 1 δ m + 3 n , (12)

where

γ 1 = [ 1 − E Z m − M ] , γ 2 = [ 120 − 56 E Z m − 8 M ] , γ 3 = [ 1191 − 245 E Z m + 19 M ] , γ 4 = [ 2416 ] ,

γ 5 = [ 1191 + 245 E Z m − 19 M ] , γ 6 = [ 120 + 56 E Z m + 8 M ] , γ 7 = [ 1 + E Z m + M ] , m = 0 , 1 , ⋯ , N , E = 7 ε 2 h Δ t , M = 105 μ h 3 Δ t . (13)

The system (12) contains of ( N + 1 ) linear equations containing ( N + 7 ) unknown coefficients ( δ − 3 , δ − 2 , δ − 1 , ⋯ , δ N + 1 , δ N + 2 , δ N + 3 ) T . To acquire a solution of this system, we require six additional restrictions. These are obtained from the boundary conditions (2) and can be used to remove δ − 3 , δ − 2 , δ − 1 and δ N + 1 , δ N + 2 , δ N + 3 from the systems (12) which occurs a matrix equation for the N + 1 unknowns d n = ( δ 0 , δ 1 , ⋯ , δ N ) T of the form

A d n + 1 = B d n . (14)

The resulting system is effectively solved with a version of the Thomas algorithm and we implement an inner iteration δ n ∗ = δ n + 1 2 ( δ n − δ n − 1 ) at each time step to overcome the non-linearity caused by Z m . Before the beginning of the solution procedure, initial parameters d 0 are established by using the initial condition and following derivatives at the boundaries;

U N ( x , 0 ) = U ( x m , 0 ) ; m = 0 , 1 , 2 , ⋯ , N (15)

( U N ) x ( a , 0 ) = 0 , ( U N ) x ( b , 0 ) = 0 , (16)

( U N ) x x ( a , 0 ) = 0 , ( U N ) x x ( b , 0 ) = 0 , (17)

( U N ) x x x ( a , 0 ) = 0 , ( U N ) x x x ( b , 0 ) = 0. (18)

So, by taking account (18), we obtain the following matrix form for the initial vector d 0 ;

W d 0 = b ,

where

W = [ 1536 2712 768 24 82731 81 210568.5 81 104796 81 10063.5 81 1 9600 81 96597 81 195768 81 96474 81 120 1 ⋱ 1 120 1191 2416 1191 120 1 1 120 96474 81 195768 81 96597 81 9600 81 1 10063.5 81 104796 81 210568.5 81 82731 81 24 768 2712 1536 ]

d 0 = ( δ 0 , δ 1 , δ 2 , ⋯ , δ N − 2 , δ N − 1 , δ N ) T

and

b = ( U ( x 0 , 0 ) , U ( x 1 , 0 ) , ⋯ , U ( x N − 1 , 0 ) , U ( x N , 0 ) ) T .

To implement the von Neumann stability analysis, GKdV equation is linearized by thinking about the quantity U p in the nonlinear term U p U x is locally invariable. Substituting the Fourier mode δ m n = ξ n e i m k h , ( i = − 1 ) in which k is a mode number and h is the element size, into the Equation (12) gives the growth factor ξ of the form

ξ = A − i B A + i B (19)

where

A = 2 cos ( 3 k h ) + 240 cos ( 2 k h ) + 2382 cos ( k h ) + 2416 B = 2 ( E Z m + M ) sin ( 3 k h ) + 2 ( 56 E Z m + 8 M ) sin ( 2 k h ) + 2 ( 245 E Z m − 19 M ) sin ( k h ) (20)

The modulus of the (19) is found 1, hence the linearized algorithm is unconditionally stable.

In this section, we introduce some numerical examples including: motion of single soliton wave whose exact solution is known to test validity of our algorithm for solving GKdV equation. The initial boundary value problem (1)-(2) possesses following conservative quantities;

I 1 = ∫ − ∞ ∞ U ( x , t ) d x , I 2 = ∫ − ∞ ∞ U 2 ( x , t ) d x , I 3 = ∫ − ∞ ∞ [ U p + 2 ( x , t ) − μ ( p + 1 ) ( p + 2 ) 2 ε U x 2 ( x , t ) ] d x (21)

which correspond to the mass, momentum and energy of the shallow water waves, respectively [

L 2 = ‖ U e x a c t − U N ‖ 2 ≃ h ∑ j = 0 N | U j e x a c t − ( U N ) j | 2 ,

and

L ∞ = ‖ U e x a c t − U N ‖ ∞ ≃ max j | U j e x a c t − ( U N ) j | .

The Motion of Single Solitary WaveFor this test problem, Equation (1) is examined with the boundary conditions U → 0 as x → ± ∞ and the initial condition U ( x , 0 ) = A sech 2 p [ k ( x − x 0 ) ] where A = [ c ( p + 1 ) ( p + 2 ) 2 ε ] 1 p is amplitude and k = p 2 c μ is width of the single soliton. The exact soliton solution of the GKdV equation is

U ( x , t ) = A sech 2 p [ k ( x − x 0 − c t ) ] ,

where c and x 0 are arbitrary constants. In order to exemplify the validity of our numerical algorithm, we conceive the first case of a single soliton solution for the parameters p = 1 , ε = 1 , μ = 4.84 × 10 − 4 , h = 0.01 , Δ t = 0.005 , c = 0.3 and ε = 3 , μ = 1 , h = 0.1 , Δ t = 0.01 through the interval [ 0,80 ] to compare with that of previous papers [

Solitary wave profiles are demonstrated at t = 0 , 0.1 , 0.2 , ⋯ , 1 in

Method | Time | I 1 | I 2 | I 3 | L 2 × 10 3 | L ∞ × 10 3 |
---|---|---|---|---|---|---|

μ = 4.84 × 10 − 4 Present Method | 0.00 | 0.144598 | 0.086759 | 0.046850 | 0 | 0 |

0.25 | 0.144598 | 0.086759 | 0.046850 | 0.02315 | 0.06802 | |

0.50 | 0.144598 | 0.086759 | 0.046850 | 0.04525 | 0.12487 | |

0.75 | 0.144598 | 0.086759 | 0.046850 | 0.06683 | 0.18353 | |

1.00 | 0.144593 | 0.086759 | 0.046850 | 0.09082 | 0.23617 | |

[ | 1.00 | 0.144598 | 0.086759 | 0.046850 | 0.13010 | 0.36895 |

[ | 1.00 | 0.144592 | 0.086759 | 0.016870 | 22.2 | |

[ | 1.00 | 0.144606 | 0.086759 | 0.046850 | 0.062 | 0.133 |

[ | 1.00 | 0.144623 | 0.086765 | 0.046847 | 2.751 | 5.018 |

[ | 1.00 | 0.144598 | 0.086759 | 0.046849 | 1.013 | 2.090 |

[ | 1.00 | 0.144261 | 0.086762 | 0.046842 | 2.606 | 6.345 |

[ | 1.00 | 0.144601 | 0.086760 | 0.046850 | 0.046 | 0.136 |

[ | 1.00 | 0.144599 | 0.086759 | 0.046850 | 0.079 | 0.238 |

μ = 1 Present Method | 0.00 | 2.190844 | 0.438176 | 0.078871 | 0 | 0 |

0.25 | 2.190844 | 0.438176 | 0.078871 | 0.038 | 0.051 | |

0.50 | 2.190858 | 0.438176 | 0.078871 | 0.040 | 0.048 | |

0.75 | 2.190848 | 0.438176 | 0.078871 | 0.059 | 0.113 | |

1.00 | 2.190873 | 0.438176 | 0.078863 | 0.083 | 0.155 |

For the second set, we choose the parameters p = 2 , ε = 3 , μ = 1 , h = 0.1 , Δ t = 0.01 , c = 0.845 and 0.3 throughout the interval [ 0,80 ] just to be able to compare them with earlier papers [

Finally, we have taken the parameters p = 3 , ε = 1 , μ = 4.84 × 10 − 4 , h = 0.01 , Δ t = 0.005 , c = 0.3 and ε = 6 , μ = 1 , h = 0.1 , Δ t = 0.01 , c = 0.6 over the region [ 0,80 ] . Thereby, solitary wave has amplitude 1.44 and 1.0, respectively. Simulations are executed to time t = 1 to invent the error norms L 2 and L ∞ and the numerical invariants I 1 , I 2 and I 3 . The calculated values are presented in

In this paper, a septic B-spline collocation method has been successfully applied to the GKdV equation to examine the motion of a single solitary wave whose analytical solution is known. To show how good and accurate the numerical solutions of the test problems, we have computed the error norms L 2 and L ∞ and conserved quantities I 1 , I 2 and I 3 . According to the tables in the paper, one can have easily seen that our error norms are enough small and the invariants are well conserved. Stability analysis has been done and the linearized numerical scheme has been obtained unconditionally stable. Thus, we can say

Method | Time | I 1 | I 2 | I 3 | L 2 × 10 3 | L ∞ × 10 3 |
---|---|---|---|---|---|---|

c = 0.845 Present Method | 0 | 4.442865 | 3.676941 | 2.071335 | 0 | 0 |

5 | 4.442865 | 3.676941 | 2.073758 | 0.917706 | 0.562852 | |

10 | 4.442865 | 3.676941 | 2.073900 | 1.265494 | 0.850150 | |

15 | 4.442865 | 3.676941 | 2.073930 | 1.638275 | 1.096426 | |

20 | 4.442865 | 3.676941 | 2.073948 | 1.983089 | 1.309575 | |

[ | 20 | 4.442866 | 3.676941 | 2.073841 | 3.656694 | 2.294197 |

[ | 20 | 4.442866 | 3.676941 | 2.073846 | 3.641638 | 2.285638 |

c = 0.3 Present Method | 0.00 | 4.442815 | 2.190881 | 0.438173 | 0 | 0 |

0.25 | 4.442818 | 2.190881 | 0.438179 | 0.046 | 0.030 | |

0.50 | 4.442818 | 2.190881 | 0.438191 | 0.073 | 0.042 | |

0.75 | 4.442818 | 2.190881 | 0.438202 | 0.091 | 0.048 | |

1.00 | 4.442818 | 2.190881 | 0.438213 | 0.105 | 0.051 | |

[ | 1.00 | 4.44192 | 2.18994 | 0.437763 | - | 0.310 |

[ | 1.00 | 4.44198 | 2.18974 | 0.437642 | - | 0.325 |

Method | Time | I 1 | I 2 | I 3 | L 2 × 10 3 | L ∞ × 10 3 |
---|---|---|---|---|---|---|

c = 0.845 Present Method | 0 | 0.162456 | 0.144101 | 0.061758 | 0 | 0 |

0.25 | 0.162456 | 0.144100 | 0.061756 | 0.443 | 1.541 | |

0.5 | 0.162458 | 0.144100 | 0.061755 | 0.926 | 3.062 | |

0.75 | 0.162456 | 0.144099 | 0.061753 | 1.456 | 4.867 | |

1.00 | 0.162450 | 0.144099 | 0.061752 | 2.008 | 6.645 | |

c = 0.3 Present Method | 0.00 | 3.620369 | 2.226620 | 0.318081 | 0 | 0 |

0.25 | 3.620352 | 2.226620 | 0.318081 | 0.026 | 0.034 | |

0.50 | 3.620352 | 2.226620 | 0.318080 | 0.029 | 0.031 | |

0.75 | 3.620334 | 2.226620 | 0.318079 | 0.043 | 0.076 | |

1.00 | 3.620342 | 2.226620 | 0.318075 | 0.062 | 0.111 |

that our numerical scheme is useful to obtain the numerical solutions of other important nonlinear problems in various fields.

The author declares no conflicts of interest regarding the publication of this paper.

Geyikli, T. (2020) Collocation Method for Solving the Generalized KdV Equation. Journal of Applied Mathematics and Physics, 8, 1123-1134. https://doi.org/10.4236/jamp.2020.86085