Startseite New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation
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New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation

  • Ahmed Hussein Msmali , Mohammad Tamsir , Neeraj Dhiman EMAIL logo und Mohammed A. Aiyashi
Veröffentlicht/Copyright: 6. Dezember 2021
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Aus der Zeitschrift Open Physics Band 19 Heft 1

Abstract

The objective of this work is to propose a collocation technique based on new cubic trigonometric B-spline (NCTB-spline) functions to approximate the regularized long-wave (RLW) equation. This equation is used for modelling numerous problems occurring in applied sciences. The NCTB-spline collocation method is used to integrate the spatial derivatives. We use the Rubin–Graves linearization technique to linearize the non-linear term. The accuracy and efficiency of the technique are examined by employing it on three important numerical examples which have three invariants of motion viz. mass, momentum, and energy. It is observed that the error norms of the present method are less than the error norms of the methods available in the literature. The numerical values of these invariants have also been approximated, which remain conserved during the program run which shows that the propagation of the solitary wave is represented perfectly. The propagation of one and two solitary waves and undulations of waves are depicted graphically. The stability analysis shows that the method is unconditionally stable.

Keywords: RLW equation; NCTB-spline functions; Rubin–Graves-type linearization technique; collocation technique; stability analysis

1 Introduction

The regularized long-wave (RLW) equation is used as a model to describe long-wave behavior. Peregrine [1] introduced this equation to demonstrate the development of undular bore. Benjamin [2] discussed the properties of this equation. The aforesaid equation has been used to model various phenomena in science such as hydromagnetic waves, longitudinal dispersive waves, shallow water waves, anharmonic lattice, and rotating flow down a tube. Olver [3] proved that the aforesaid equation has only three conservation laws. The fixed conservation laws show that this equation is non-integrable. In fact, there are various initial and boundary conditions for that the general solution has not been obtained. Hence, there are only a few analytical solutions with a limited set of these conditions. So, there is a need for numerical methods for the RLW equation with different sets of initial and boundary conditions.

Various approximations have been found in the literature for the RLW equation with several sets of initial and boundary conditions. For instance, Gardner et al. [4] solved it numerically by the least square method based on linear space-time finite elements, whereas Dogan [5] used linear finite element-based Galerkin method. The authors of Ref. [6] used the splitting method in combination with the CB-spline method for solving non-linear RLW equation. The authors of Refs. [7,8] proposed linearized implicit and three-level finite difference methods, respectively, while a new finite difference method based on the quintic spline and splitting method has been proposed by Raslan [9]. Petrov-Galerkin finite element method, least-square quadratic finite element method and least-square cubic B-spline finite element method have been proposed by authors of Refs. [10,11,12], respectively, for solving the aforesaid equation. Saka et al. [13] and Zaki [14] used splitting methods in the combination of cubic and quadratic B-spline finite elements, respectively. The collocation method based on both quadratic and quintic B-splines has been proposed by Dağ et al. [15]. Mei and Chen [16] presented explicit multistep Galerkin method to solve this equation, whereas Dag and Dereli [17] investigated it numerically using radial basis-based meshless method. Eilbeck and McGuire [18] studied the solitary wave solutions numerically. Additionally, Cimpoiasu [19] investigated some traveling wave solutions for the long–short wave resonance model.

In Ref. [20], the authors presented a differential quadrature method based on cubic B-spline basis functions, whereas Saka and Dag [21] presented a collocation method based on quartic B-spline basis functions for the numerical solution of the aforesaid equation. The authors of Refs. [22,23,24,25] used Galerkin’s method using linear finite elements, quadratic B-spline finite element-based lumped Galerkin method, B-spline finite element method, and quadratic B-spline Galerkin finite element method in combination with space-splitting technique, respectively, to investigate the motion of a single solitary wave, development of two solitary wave interaction and an undular bore, numerically. The authors of Ref. [26] proposed a fourth-order collocation method based on cubic B-splines to solve RLW and modified RLW equations. Guo et al. [27] applied multiple integral FVM based on Lagrange interpolation for the Rosenau-RLW equation. Recently, Dhiman and Tamsir [28] applied a trigonometric cubic B-spline collocation technique for Fisher s reaction-diffusion equation.

In this article, we consider the RLW equation as follows:

(1) ( u − μ u x x ) t + ( ε u + α ) u x = 0 , x ∈ [ a , b ] , t ∈ ( 0 , T ] ,

subject to

(2) u ( x , 0 ) = φ ( x ) , x ∈ [ a , b ] ,

and the boundary conditions

(3) u ( a , t ) = β 1 , u ( b , t ) = β 2 , u x ( a , t ) = 0 , u x ( b , t ) = 0 , t ∈ ( 0 , T ] ,

where μ , ε , α are known positive parameters and the wave amplitude u = u ( x , t ) is to be obtained. The terms β 1 and β 2 are given constants. Bona and Byrant [29] already proved that there exists a unique solution for the problem (1)–(3). The problem (1)–(3) conserve mass, momentum, and energy. These terms are physical quantities of motion, which are given by

(4) I 1 = ∫ a b u d x , I 2 = ∫ a b ( u 2 + μ u x 2 ) d x , I 3 = ∫ a b ( u 3 + 3 u x 3 ) d x .

The rest of the article is organized as follows. The procedure of the new trigonometric B-spline collocation technique is given in Section 2. In Section 3, a brief discretization of the problems using the proposed technique is given. The von Neumann stability is discussed in Section 4. In Section 5, numerical results are given. Finally, Section 6 focuses on the conclusions.

2 New cubic trigonometric B-spline (NCTB-spline) collocation technique

In this section, we introduce the collocation method based on NCTB-spline functions. First, we partition the problem domain x ∈ [ a , b ] into mesh of equal length h = x i + 1 − x i for i = 0 , 1 , … , M . For discrete form, we use u ( x i , t j ) = u i j for i = 0 , 1 , … , M and j = 0 , 1 , … , N . The NCTB-spline functions NB i ( x ) for i = − 1 , 0 , … , M , M + 1 are given by Ref. [30]

(5) NB i ( x ) = d i − 2 ( g 3 ( z i − 2 ) ) , x ∈ [ x i − 2 , x i − 1 ) , ∑ k = 0 3 c i − 1 , k g k ( z i − 1 ) , x ∈ [ x i − 1 , x i ) , ∑ k = 0 3 b i , k g k ( z i − 1 ) , x ∈ [ x i , x i + 1 ) , a i + 1 g 0 ( z i − 1 ) , x ∈ [ x i + 1 , x i + 2 ) , 0 , else ,

where

z i ( x ) = π 2 x − x i x i + 1 − x i , i = 1 , 2 , … , a i = b i , 3 = c i , 0 = d i = v l i 2 μ i ψ i , b i , 0 = v ψ i ( α i l i + β i l i − 1 ) μ i ψ i 2 , b i , 1 = v β i ψ i , b i , 2 = v l i + 1 ψ i , c i , 1 = v l i − 1 ψ i , c i , 2 = v α i ψ i , c i , 3 = v ψ i ( α i l i + 1 + β i l i ) μ i ψ i 2 ,

where

l i = x i + 1 − x i , μ i = l i − 1 + l i , α i = l i − 1 + v l i , β i = v l i − 1 + l i , v = 1 + 2 η , ψ i = v l i − 2 + v 2 l i − 1 + v l i .

The new trigonometric functions are defined as:

(6) g 0 ( z ) = ( 1 − sin z ) 2 ( 1 + ( 1 − η ) sin z ) , g 1 ( z ) = sin z ( 1 − sin z ) ( 2 − ( 1 − η ) ( 1 − sin z ) ) , g 2 ( z ) = cos z ( 1 − cos z ) ( 2 − ( 1 − η ) ( 1 − cos z ) ) , g 3 ( z ) = ( 1 − cos z ) 2 ( 1 + ( 1 − η ) cos z ) ,

where η ∈ 1 2 , 2 is the shape parameter. The set of NCTB-spline basis functions { NB i ( x ) : i = − 1 , 0 , … , M , M + 1 } forms a basis over the problem domain. Table 1 gives the values of NB i ( x ) and it is two derivatives at the knots.

Table 1

Values of NB i ( x ) , NB i ′ ( x ) , and NB i ″ ( x ) at the knots

x i − 2 x i − 1 x i x i + 1 x i + 2
NB i ( x ) 0 ξ 1 ξ 2 ξ 3 0
N B i ′ ( x ) 0 ξ 4 ξ 5 ξ 6 0
NB i ″ ( x ) 0 ξ 7 ξ 8 ξ 9 0

We assume that the approximation u M at ( x i , t j ) is written as a linear combination of the NCTB-spline functions and unknown time-dependent coefficients as:

(7) u M ( x , t ) = ∑ m = − 1 M + 1 C m ( t ) NB m ( x ) .

The NCTB-spline and its four principle derivatives vanish outside of the region [ x i − 2 , x i + 2 ] . Since each NCTB-spline covers four elements, each element is covered by four NCTB-splines. Hence, principle four NCTB-splines cover the interval [ x i , x i + 1 ]. So, the variation of the U M ( x , t ) , over the element, can be expressed as

(8) u M ( x , t ) = ∑ m = i − 1 i + 1 C m ( t ) NB m ( x ) .

From Eq. (8) and Table 1, we get the approximated u ( x , t ) , u x ( x , t ) , and u x x ( x , t ) as follows:

(9) u i j = ξ 1 C i − 1 j + ξ 2 C i j + ξ 3 C i + 1 j ,

(10) ( u x ) i j = ξ 4 C i − 1 j + ξ 5 C i j + ξ 6 C i + 1 j ,

(11) ( u x x ) i j = ξ 7 C i − 1 j + ξ 8 C i j + ξ 9 C i + 1 j ,

where C i j = C i ( t j ) .

3 Discretization of the RLW equation

This section considers the problem (1)–(3) when μ , ε , α are given. First, we discretize Eq. (1) as follows:

(12) ( u − μ ( u x x ) ) i j + 1 − ( u − μ ( u x x ) ) i j Δ t + ε 2 ( ( u u x ) i j + 1 + ( u u x ) i j ) + α 2 ( ( u x ) i j + 1 + ( u x ) i j ) = 0 .

We linearize the non-linear term by Rubin–Graves technique [31] as follows:

(13) u j + 1 ( u x ) j + 1 = u j ( u x ) j + 1 + u j + 1 ( u x ) j − u j ( u x ) j .

Using Eq. (13) in Eq. (12), we get

(14) ( u − μ ( u x x ) ) i j + 1 + Δ t ε 2 ( u i j ( u x ) i j + 1 + u i j + 1 ( u x ) i j ) + Δ t α 2 ( u x ) i j + 1 = ( u − μ ( u x x ) ) i j − Δ t α 2 ( u x ) i j .

Simplifying the above equation, we get

(15) P i j u i j + 1 + Q i j ( u x ) i j + 1 − μ ( u x x ) i j + 1 = u i j − Δ t α 2 ( u x ) i j − μ ( u x x ) i j ,

where

P i j = 1 + Δ t ε 2 ( u x ) i j , Q i j = Δ t ε 2 u i j + Δ t α 2 .

Now, using equations (9)–(11), we get

(16) ( P i j ξ 1 + Q i j ξ 4 − μ ξ 7 ) C i − 1 j + 1 + ( P i j ξ 2 + Q i j ξ 5 − μ ξ 8 ) C i j + 1 + ( P i j ξ 3 + Q i j ξ 6 − μ ξ 9 ) C i + 1 j + 1 = ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 C i − 1 j + ξ 2 − Δ t α 2 ξ 5 − μ ξ 8 C i j + ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 C i + 1 j , i = 1 , 2 , … , M − 1 , j = 0 , 1 , … , N .

Now, we discretize the boundary conditions as follows:

(17) ξ 1 C − 1 j + ξ 2 C 0 j + ξ 3 C 1 j = β 1 , j = 1 , 2 , … , N ,

(18) ξ 1 C M − 1 j + 1 + ξ 2 C M j + 1 + ξ 3 C M + 1 j + 1 = β 2 , j = 1 , 2 , … , N .

From equations (17) and (18), we get

(19) C − 1 j = β 1 ξ 1 − ξ 2 ξ 1 C 0 j − ξ 3 ξ 1 C 1 j , j = 1 , 2 , … , N ,

(20) C M + 1 j = β 2 ξ 3 − ξ 1 ξ 3 C M − 1 j − ξ 2 ξ 3 C M j , j = 1 , 2 , … , N .

For i = 0 , using (19) in (16), we get

(21) P 0 j ξ 2 + Q 0 j ξ 5 − μ ξ 8 − ξ 2 ξ 1 ( P 0 j ξ 1 + Q 0 j ξ 4 − μ ξ 7 ) C 0 j + 1 + ( P 0 j ξ 3 + Q 0 j ξ 6 − μ ξ 9 − ξ 3 ξ 1 ( P 0 j ξ 1 + Q 0 j ξ 4 − μ ξ 7 ) ) C 1 j + 1 = ξ 2 − Δ t α 2 ξ 5 − μ ξ 8 − ξ 2 ξ 1 ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 C 0 j + ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 − ξ 3 ξ 1 ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 C 1 j + α 1 ξ 1 ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 − α 1 ξ 1 ( P 0 j ξ 1 + Q 0 j ξ 4 − μ ξ 7 ) ) , j = 0 , 1 , … , N .

For i = M , using (20) in (16), we get

(22) ( P M j ξ 1 + Q M j ξ 4 − μ ξ 7 ) − ξ 1 ξ 3 ( P M j ξ 3 + Q M j ξ 6 − μ ξ 9 ) C M − 1 j + 1 + ( ( P M j ξ 2 + Q M j ξ 5 − μ ξ 8 ) − ξ 2 ξ 3 ( P M j ξ 3 + Q M j ξ 6 − μ ξ 9 ) ) C M j + 1 = ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 − ξ 1 ξ 3 ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 C M − 1 j + ξ 2 − Δ t α 2 ξ 5 − μ ξ 8 − ξ 2 ξ 3 ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 C M j + α 2 ξ 3 ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 − ( P M j ξ 3 + Q M j ξ 6 − μ ξ 9 ) , j = 0 , 1 , … , N .

At t j + 1 , j = 0 , 1 , … , N , Eqs. (21), (16), and (22) can be written into the form of a linear system of order ( M + 1 ) × ( M + 1 ) as follows:

q ˆ 0 j − ξ 2 ξ 1 p ˆ 0 j r ˆ 0 j − ξ 3 ξ 1 p ˆ 0 j p ˆ 1 j q ˆ 1 j r ˆ 1 j p ˆ 2 j q ˆ 2 j r ˆ 2 j ⋱ ⋱ ⋱ p ˆ M − 2 j q ˆ M − 2 j r ˆ M − 2 j p ˆ M j − ξ 1 ξ 3 r ˆ M j q ˆ M j − ξ 2 ξ 3 r ˆ M j C 0 j + 1 C 1 j + 1 C 2 j + 1 ⋮ C M − 2 j + 1 C M − 1 j + 1 C M j + 1 = R 0 j R 1 j R 2 j ⋮ R M − 2 j R M − 1 j R M j ,

where

p ˆ i j = ( P i j ξ 1 + Q i j ξ 4 − μ ξ 7 ) , q ˆ i j = ( P i j ξ 2 + Q i j ξ 5 − μ ξ 8 ) , r ˆ i j = ( P i j ξ 3 + Q i j ξ 6 − μ ξ 9 ) , R 0 j = ξ 2 − Δ t α 2 ξ 5 − μ ξ 8 − ξ 2 ξ 1 ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 C 0 j + ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 − ξ 3 ξ 1 ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 C 1 j + α 1 ξ 1 ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 − ( P 0 j ξ 1 + Q 0 j ξ 4 − μ ξ 7 ) , j = 0 , 2 , … , N , R i j ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 C i − 1 j + ξ 2 − Δ t α 2 ξ 5 − μ ξ 8 C i j + ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 C i + 1 j , i = 1 , 2 , … , M − 1 , j = 0 , 1 , … , N , R M j = ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 − ξ 1 ξ 3 ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 C M − 1 j + ξ 2 − Δ t α 2 ξ 5 − μ ξ 8 − ξ 2 ξ 3 ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 C M j + α 2 ξ 3 ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 − ( P M j ξ 3 + Q M j ξ 6 − μ ξ 9 ) , j = 0 , 1 , … , N .

Now, we find the vector ( C − 1 0 , C 0 0 , … , C M 0 , C M + 1 0 ) from the initial condition which gives M + 1 equation in N + 3 unknowns. To eliminate the unknowns C − 1 0 and C M + 1 0 , we use u x ( a , t ) = 0 and u x ( b , t ) = 0 , which gives

(23) C − 1 0 = − ξ 5 ξ 4 C 0 0 − ξ 6 ξ 4 C 1 0 ,

(24) C M + 1 0 = − ξ 4 ξ 6 C M − 1 0 − ξ 5 ξ 6 C M 0 .

Removing C − 1 0 and C M + 1 0 from the system, we have a tridiagonal system of order ( M + 1 ) × ( M + 1 ) as follows:

ξ 2 − ξ 1 ξ 5 ξ 4 ξ 3 − ξ 1 ξ 6 ξ 4 ξ 1 ξ 2 ξ 3 ξ 1 ξ 2 ξ 3 ⋱ ⋱ ⋱ ξ 1 ξ 2 ξ 3 ξ 1 − ξ 1 ξ 4 ξ 6 ξ 2 − ξ 1 ξ 5 ξ 6 C 0 0 C 1 0 C 2 0 ⋮ C M − 1 0 C M 0 = φ ( x 0 ) − ξ 1 ξ 4 φ ′ ( x 0 ) φ ( x 1 ) φ ( x 2 ) ⋮ φ ( x M − 1 ) φ ( x M ) − ξ 1 ξ 6 φ ′ ( x M ) .

4 Stability analysis

Now, the von-Neumann stability analysis [32,33,34,35,36] is carried out for the RLW equation. We consider the discretized system (12) as

(25) ( u − μ ( u x x ) ) i j + 1 + Δ t ε 2 ( u i j ( u x ) i j + 1 + u i j + 1 ( u x ) i j ) + Δ t α 2 ( u x ) i j + 1 = ( u − μ ( u x x ) ) i j − Δ t α 2 ( u x ) i j .

Using equations (9)–(11), we get

(26) 1 + Δ t ε 2 ( u x ) i j ξ 1 + Δ t ε 2 u i j + Δ t α 2 ξ 4 − μ ξ 7 C i − 1 j + 1 + 1 + Δ t ε 2 ( u x ) i j ξ 2 + Δ t ε 2 u i j + Δ t α 2 ξ 5 − μ ξ 8 C i j + 1 + 1 + Δ t ε 2 ( u x ) i j ξ 3 + Δ t ε 2 u i j + Δ t α 2 ξ 6 − μ ξ 9 C i + 1 j + 1 = ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 C i − 1 j + ξ 2 − Δ t α 2 ξ 5 − μ ξ 8 C i j + ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 C i + 1 j .

Now, we assume u = k ¯ 1 and u x = k ¯ 2 as local constants for known level to get

(27) ( A ˆ ξ 1 + B ˆ ξ 4 − μ ξ 7 ) C i − 1 j + 1 + ( A ˆ ξ 2 + B ˆ ξ 5 − μ ξ 8 ) C i j + 1 + ( A ˆ ξ 3 + B ˆ ξ 6 − μ ξ 9 ) C i + 1 j + 1 = ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 C i − 1 j + ξ 2 − Δ t α 2 ξ 5 − μ ξ 8 C i j + ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 C i + 1 j ,

where

A ˆ = 1 + Δ t ε 2 k 2 , B ˆ = Δ t ε 2 k 1 + Δ t α 2 .

Next, we consider C i j = δ j e k i ϕ as trial solutions at a given point x i , where ϕ = θ h and k = − 1 . The θ is the mode number and h is the element size. Substituting the trial solution in the above equation, we get

(28) ( A ˆ 1 cos ( k ϕ ) + A ˆ 2 + A ˆ 3 cos ( k ϕ ) − i ( A ˆ 1 + A ˆ 3 ) sin ( k ϕ ) ) δ B ˆ = B ˆ 1 cos ( k ϕ ) + B ˆ 2 + B ˆ 3 cos ( k ϕ ) − i ( B ˆ 1 + B ˆ 3 ) sin ( k ϕ ) ,

where

A ˆ 1 = A ˆ ξ 1 + B ˆ ξ 4 − μ ξ 7 , A ˆ 2 = A ˆ ξ 2 + B ˆ ξ 5 − μ ξ 8 , A ˆ 3 = A ˆ ξ 3 + B ˆ ξ 6 − μ ξ 9 , B ˆ 1 = ξ 1 − Δ t α 2 ξ 4 − μ ξ 7 , B ˆ 2 = ξ 2 − Δ t α 2 ξ 5 − μ ξ 8 , B ˆ 3 = ξ 3 − Δ t α 2 ξ 6 − μ ξ 9 .

Now, simplifying the terms in equation (28), we get

(29) ∣ δ ∣ 2 = ( B ˆ 1 − B ˆ 2 + B ˆ 3 ) 2 + 4 ( B ˆ 1 + B ˆ 3 ) B ˆ 2 cos 2 ( k ϕ ) ( A ˆ 1 − A ˆ 2 + A ˆ 3 ) 2 + 4 ( A ˆ 1 + A 3 ) A ˆ 2 cos 2 ( k ϕ ) ,

where

A ˆ 1 − A ˆ 2 + A ˆ 3 = ( ξ 1 − ξ 2 + ξ 3 ) − Δ t α 2 ( ξ 4 − ξ 5 + ξ 6 ) − μ ( ξ 7 − ξ 8 + ξ 9 ) + Δ t α 2 ( k 2 ( ξ 1 − ξ 2 + ξ 3 ) + α ( k 1 + 2 ) ( ξ 4 − ξ 5 + ξ 6 ) ) , B ˆ 1 − B ˆ 2 + B ˆ 3 = ( ξ 1 − ξ 2 + ξ 3 ) − Δ t α 2 ( ξ 4 − ξ 5 + ξ 6 ) − μ ( ξ 7 − ξ 8 + ξ 9 ) .

Since ξ 1 − ξ 2 + ξ 3 > 0 and ξ 4 − ξ 5 + ξ 6 > 0 , we get ( A ˆ 1 − A ˆ 2 + A ˆ 3 ) 2 ≥ ( B ˆ 1 − B ˆ 2 + B ˆ 3 ) 2 . Also, A ˆ 1 ≥ B ˆ 1 , A ˆ 2 ≥ B ˆ 2 , and A ˆ 3 ≥ B ˆ 3 . Therefore, we get ∣ δ ∣ ≤ 1 , and hence the method is unconditionally stable for the discretized system of the RLW equation.

5 Results and discussion

In this section, we consider three numerical examples in order to check the accuracy and efficiency of the proposed method. The L 2 and L ∞ errors are given by

(30) L 2 = h ∑ i = 0 M ∣ u i − u i e ∣ 2 1 / 2 , L ∞ = max 0 ≤ i ≤ M ∣ u i − u i e ∣ ,

where u i and u e are numerical and analytical solutions at x i .

5.1 Example 1

First, we consider the RLW equation (1) in x ∈ [ − 40 , 60 ] with analytical solution [8,26,37]

(31) u ( x , t ) = 3 c sech 2 ( k x − k v t − k x 0 ) ,

and the boundary conditions (3), where β 1 = β 2 = 0 . This exact solution corresponds to the single solitary wave centrally located at x 0 initially with amplitude 3 c and with k , where k = 1 2 c ε μ v 1 2 . The term v = 1 + c ε stands for the wave velocity. The exact values of mass, momentum, and energy are given by

(32) I 1 = 6 c k , I 2 = 12 c 2 k + 48 k c 2 μ 5 , I 3 = 36 c 2 k + 144 c 3 5 k .

We fix ε = 1 , μ = 1 , α = 1 , and x 0 = 0 to approximate Example 1. First, we approximate the error norms for Example 1 with parameters M = 200 , Δ t = 0.001 for x = c = 0.1 and 0.3 at different t , which are presented in Table 2. It is clear from this table that the obtained results are better than the results presented by Mittal and Rohila [26]. Next, we computed the L ∞ error norms with M = 200 , Δ t = 0.1 , t = 20 , which are presented in Tables 3 and 4 for c = 0.1 and 0.3, respectively, together with the exact values of mass, energy, and momentum. We found from these tables that the present results are better than the results obtained by Dag and Ozer [12], Jain et al. [6], Kutluay and Esen [8], Mittal and Rohila [26], Raslan [9], two methods of Saka and Dag [21] viz. QBCM1 and QBCM2, and Zaki [14]. Moreover, during the program run, we observe that the approximated values of mass, energy, and momentum remain almost constants, i.e., propagation of the solitary wave represented accurately. The comparison between numerical and analytical solutions together with absolute error norms for M = 200 , Δ t = 0.1 for c = 0.1 at t = 4 , 8, and 16 are depicted in Figure 1. One can see from these figures that there is an excellent agreement between approximate and analytical solutions. Figure 2 shows the profiles of single solitary wave at various time with M = 200 , Δ t = 0.1 for c = 0.1 and 0.3, respectively, at t = 1 , 5 , 10 , 15 , and 20.

Table 2

Comparison of the present solutions with the solutions obtained by Mittal and Rohila [26], in terms of L ∞ , for Example 1

c = 0.1 c = 0.3
t Mittal and Rohila [26] Present Mittal and Rohila [26] Present
1 4.57 × 1 0 7 2.01 × 1 0 7 1.00 × 1 0 5 5.40 × 1 0 6
2 3.79 × 1 0 5 1.83 × 1 0 7 1.84 × 1 0 5 5.00 × 1 0 6
3 3.80 × 1 0 5 1.01 × 1 0 7 6.09 × 1 0 5 5.50 × 1 0 6
4 3.80 × 1 0 5 1.06 × 1 0 7 6.08 × 1 0 5 5.25 × 1 0 6
5 2.23 × 1 0 6 1.90 × 1 0 7 6.08 × 1 0 5 4.57 × 1 0 6
6 2.75 × 1 0 6 1.07 × 1 0 7 6.08 × 1 0 5 3.10 × 1 0 6
7 3.14 × 1 0 6 1.05 × 1 0 7 6.09 × 1 0 5 2.25 × 1 0 6
8 3.00 × 1 0 6 1.54 × 1 0 7 6.09 × 1 0 5 2.12 × 1 0 6
9 3.82 × 1 0 6 1.50 × 1 0 7 6.08 × 1 0 5 1.80 × 1 0 6
10 3.78 × 1 0 6 1.46 × 1 0 7 6.09 × 1 0 5 1.35 × 1 0 6
Table 3

Comparison of the present method and existing method results with c = 0.03 , for Example 1

Methods L ∞ I 1 I 2 I 3
Analytical — 3.97992 0.81046 2.57900
Present method 1.12 × 1 0 6 3.97993 0.81046 2.57901
Dag and Ozer [12] 1.50 × 1 0 3 3.96466 0.80461 2.56971
Jain et al. [6] 6.70 × 1 0 2 4.41219 0.89734 2.85361
Kutluay and Esen [8] 2.10 × 1 0 4 3.97997 0.81045 2.57901
Mittal and Rohila [26] 7.81 × 1 0 5 3.97993 0.81046 2.57901
Raslan [9] 2.27 × 1 0 4 3.97995 0.80972 2.57607
QBCM1 [21] 8.30 × 1 0 5 3.97995 0.81046 2.57901
QBCM2 [21] 1.29 × 1 0 4 3.97995 0.81046 2.57901
Zaki [14] 1.81 × 1 0 4 3.97781 0.80963 2.5762
Table 4

L 2 and L ∞ error norms with c = 0.1 , Δ t = 0.1 , and Δ x = 0.125 , for Example 1

t QBSCM L 2 × 1 0 4 QBSCM L ∞ × 1 0 4 Present method L 2 × 1 0 4 Present method L ∞ × 1 0 4
0 0 0 0 0
4 0.1757 0.0693 0.1258 0.0548
8 0.2249 0.0887 0.1565 0.0652
12 0.3355 0.1072 0.2547 0.0923
16 0.4075 0.1224 0.3582 0.1567
20 0.4315 0.1321 0.3649 0.1825
Figure 1 Numerical and analytical solutions together with absolute error norms at: (a) t = 4 t=4 , (b) t = 8 t=8 , and (c) t = 16 t=16 , for Example 1.
Figure 1

Numerical and analytical solutions together with absolute error norms at: (a) t = 4 , (b) t = 8 , and (c) t = 16 , for Example 1.

Figure 2 Single solitary wave profiles with M = 200 M=200 for: (a) c = 0.1 c=0.1 and (b) c = 0.3 c=0.3 , for Example 1.
Figure 2

Single solitary wave profiles with M = 200 for: (a) c = 0.1 and (b) c = 0.3 , for Example 1.

5.2 Example 2

Now, we consider the RLW (1) in interval x ∈ [ 0 , 120 ] with the initial condition [8,9,26,37]

(33) u ( x , 0 ) = ∑ j = 1 2 3 C j sech 2 ( k j ( x − x j ) ) ,

where C j = 4 k j 2 1 − 4 k j 2 , j = 1 , 2 . The RLW Eq. (1) with this initial condition represents two positive solitary wave equations. For the numerical computation of this example, we choose k 1 = 0.4 , k 2 = 0.3 , x 1 = 15 , x 2 = 35 , ε = 1 , α = 1 , and μ = 1 . Figure 3 depicts the propagation of two positive solitary waves with N = 300 for t = 0 , 5 , 10 , 15 , 20 , and 25, respectively. We also check the efficiency of the proposed method by computing the invariant mass, momentum, and energy. Tables 5 and 6 show the comparison of the computed invariants with the invariants presented in Refs. [8,26]. It is clear from these figures and tables that an excellent agreement is found between present and existing results. Also, it is noticed that the exact values of mass, energy, and momentum remain conserved during the program run which shows that the propagation of the solitary wave represented perfectly.

Figure 3 Plot of the numerical solutions for M = 300 M=300 at different time levels: (a) t = 0 t=0 , (b) t = 5 t=5 , (c) t = 10 t=10 , (d) t = 15 t=15 , (e) t = 20 t=20 , and (f) t = 25 t=25 , for Example 2.
Figure 3

Plot of the numerical solutions for M = 300 at different time levels: (a) t = 0 , (b) t = 5 , (c) t = 10 , (d) t = 15 , (e) t = 20 , and (f) t = 25 , for Example 2.

Table 5

Numerical values of mass and momentum at different time levels, for Example 2

t I 1 (Mass) Present I 2 (Momentum) Present I 1 (Mass) [26] I 2 (Momentum) [26] I 1 (Mass) [8] I 2 (Momentum) [8]
0 37.9165 120.4126 37.9165 120.5220 37.9165 120.3515
2 37.9168 120.4115 37.9169 120.5230 37.9168 120.3571
4 37.9170 120.4356 37.9170 120.5240 37.9170 120.3584
6 37.9171 120.4360 37.9171 120.5320 37.9170 120.3586
8 37.9173 120.4275 37.9173 120.5560 37.9172 120.3583
10 37.9174 120.4596 37.9174 120.5950 37.9173 120.3570
12 37.9174 120.4675 37.9174 120.5340 37.9173 120.3915
14 37.9174 120.3570 37.9174 120.2570 37.9174 120.4156
16 37.9175 120.3045 37.9175 120.1940 37.9174 120.3886
18 37.9175 120.3495 37.9175 120.3570 37.9174 120.3653
20 37.9174 120.4169 37.9175 120.4620 37.9174 120.3599
22 37.9174 120.4168 37.9175 120.4620 37.9174 120.3599
24 37.9175 120.4215 37.9175 120.5160 37.9175 120.3595
Table 6

Numerical values of energy, for Example 2

t I 3 (Energy) Present I 3 (Energy) [26] I 3 (Energy) [8]
0 744.0811 744.0810 744.0814
2 744.0760 744.0770 744.0387
4 744.0560 744.0660 744.0110
6 744.0201 744.0320 744.9796
8 743.8815 743.8910 743.8679
10 743.4185 743.4130 743.4202
12 742.3056 742.2550 742.3387
14 741.5525 741.5420 741.5781
16 742.5314 742.5810 742.4890
18 7435232 743.5710 743.4752
20 743.9081 743.9290 743.8638
22 743.9081 743.9290 743.8638
24 743.8676 743.9175 744.0037

5.3 Example 3

Finally, we consider the RLW Eq. (1) with initial condition [8,26,37,38,39]

(34) u ( x , 0 ) = 0.5 β 1 1 − tanh x − x c d ,

to study the development of the undular bore together with the boundary conditions

(35) u ( a , t ) = β 1 , u ( b , t ) = 0 .

The term d denotes the slope between the shallow and deeper water, whereas u ( x , 0 ) = 0 represents the elevation of the water above the equilibrium surface at initial time. The change in water level of magnitude u ( x , 0 ) is centered at x = x c . We choose the parameters β 1 = 0.1 , ε = 1.5 , μ = 1 , α = 1 , x c = 0 , a = − 36 , b = 300 , Δ t = 0.1 , Δ x = 0.24 , d = 2 , and 5 and we run the program until t = 250 . The initial functions at t = 0 are depicted in Figure 4 for d = 5 , 2 while the undulation profiles at t = 250 are shown in Figure 5 for d = 5 and 2, respectively. Graphs in Figure 5 represent the undular bore profiles when the gentle slope d = 5 and steep slope d = 2 , respectively, are used. A few more undulations can be seen when a steep slope, i.e., d = 0 is used as the initial condition. It is noticed that an increase in number of undulations is because of the form of the initial function. Also, we see that the magnitudes of leading undulations for the d = 5 and 2 become closer and closer for the longer run. An excellent agreement between the present and existing results [1,4,15] is found.

Figure 4 Initial profile at t = 0 t=0 for: (a) d = 5 d=5 and (b) d = 2 d=2 , for Example 3.
Figure 4

Initial profile at t = 0 for: (a) d = 5 and (b) d = 2 , for Example 3.

Figure 5 Undulation profiles at time t = 250 t=250 for: (a) d = 5 d=5 and (b) d = 2 d=2 , for Example 3.
Figure 5

Undulation profiles at time t = 250 for: (a) d = 5 and (b) d = 2 , for Example 3.

6 Conclusion

An NCTB-spline collocation technique has been proposed for the numerical computation of the RLW equation. The proposed collocation technique is used to integrate the spatial derivatives, whereas the discretization of the time derivative is done by the usual finite difference method. The numerical results have been presented in tabular form as well as graphically at different time levels. The present solutions have been compared with the solutions obtained by methods of Dag and Ozer [12], Jain et al. [6], Kutluay and Esen [8], Mittal and Rohila [26], Raslan [9], two methods of Saka and Dag [21] viz. QBCM1 and QBCM2, and Zaki [14] as well as with the exact solutions. It has been observed that the present error norms are less than the error norms presented in Refs. [6,8,9,12,14,21,26]. We noticed that the physical quantities of motion remain conserved during the program run, which shows that the propagation of the solitary wave is represented perfectly. The stability analysis of the discretized system of the RLW equation shows that the method is unconditionally stable.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-09-02
Revised: 2021-11-06
Accepted: 2021-11-08
Published Online: 2021-12-06

© 2021 Ahmed Hussein Msmali et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2021-0087
Schlagwörter für diesen Artikel
RLW equation; NCTB-spline functions; Rubin–Graves-type linearization technique; collocation technique; stability analysis
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Heruntergeladen am 16.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/phys-2021-0087/html
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