Numerical Solution of Kortweg-de Vries Equation

In this paper, we are going to derive numerical methods for solving the KdV equation using Pade approximation for space direction, trapezoidal and implicit mid-point rule in the time direction. The schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities will be used to display the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be conducted. The numerical results showed, interaction behavior is elastic and the conserved quantities are conserved which is a good indication of the reliability of the schemes under consideration.


Introduction
In this work, we consider the well known Kortweg-de Vries (KdV) equation [1] ( )  [4] a class of three level finite difference scheme for solving KdV equation is proposed firstly by Zabusky and Kruskal who discovered the concept of soliton, which can be defined as a localized waves with special interaction properties, while studying the results of a numerical computation on the KdV equation. The proposed three-level scheme which requires a starting procedure satisfies momentum and energy conservation laws. In [5], a two-level hopscotch algorithm is derived and the stability and dispersion is analyzed. The Crank-Nicolson extrapolation scheme for the KdV equation is proposed in [6]. A hybrid numerical method for KdV equation is proposed by a combination of finite difference and sinc collocation method is proposed in [7]. In [8], a quadratic B-spline Galerkin finite element method is proposed. The Galerkin method for KdV equation using a new basis of smooth piecewise cubic polynomials is developed for simulating the motions and interaction of solitary waves presented in [9]. Petrov Galerkin method [10] [11] [12] [13] [14] is used to solve KdV and KdV like equations to obtain highly accurate results. Numerical solution of KdV equation by Galerkin B-spline finite element method is presented in [15]. In [16], The conservation and convergence of two finite difference schemes for KdV equations with initial and boundary conditions derived, the first scheme is a two-level nonlinear implicit finite difference scheme which is proved to be unconditional convergent, and the second one is a three-level linearized finite difference scheme and proved to be conditionally convergent.
In this paper, we will derive numerical schemes for the KdV equation; based on the Padé approximation, fourth approximation of the space derivatives is used. The second order implicit schemes using, trapezoidal rule, implicit midpoint rule (both are implicit and unconditionally stable), and the explicit Runge-Kutta method of order four are used to approximate the time derivative.
A nonlinear penta-diagonal system is obtained. Newton's method is used to The rest of the paper is organized as follow. In Section 2, we present three different schemes using fourth order Pade approximation in space direction and second order in time direction using trapezoidal and implicit mid-point rule.
Also we implement the explicit fourth Runge-Kutta method to solve the first order differential system in time. In Section 3, the accuracy of the proposed schemes is given. In Section 4, the stability analysis of scheme 1 and scheme 2 are derived using von-Neumann stability analysis. In Section 5, we present various numerical tests which validate the accuracy and the efficiency of the proposed schemes. Finally, we conclude with a brief discussion in Section 6.

Numerical Methods
In this section, we derive a high order compact finite difference method for the initial boundary value problem (1)-(3). We first describe our solution domain and its grids. The solution domain is defined to be ( ) where m U  denotes the time derivative of U at m x ; and the boundary condi- The system in (6) can be written in a matrix vector form as where M is the penta diagonal matrix The previous differential system can be solved by Trapezoidal method is of second order, implicit scheme, and A-stable method 2) Implicit mid-point rule  (10) Implicit middle point rule is of second order, implicit scheme, and A-stable method.
The previous methods (Crank-Nicolson like schemes), are usually produce unconditionally stable method, nonlinear penta diagonal system to be solved at each time step. Explicit Runge-Kutta method of fourth order will be used as well, ing Crouts method, and the method is conditionally stable. The details of the proposed schemes will be discussed in the following subsections.

Scheme 1 (Trapezoidal Rule)
Trapezoidal rule method (9) which is of second order accuracy and A-stable is used to solve the ordinary differential system in (7), we assume n m U to be the fully discrete approximation of the exact solution n m u this will lead to the non- The system in (11) is a nonlinear penta-diagonal system in the unknown vector 1 n+ U . The solution of this system can be obtained by many methods, like: Newton's method; the Predictor-Corrector method and linearization techniques.
Newton's method will be adopted in this work.

Scheme 2 (Implicit Mid-Point Rule)
A second finite difference scheme obtained by using (10) the mid-point rule which is of second order and A-stable, using this will lead us to the nonlinear scheme of the form The system in (10) can be given by the finite difference equation.
The system in (13) is a nonlinear penta-diagonal system which can be solved by many iterative methods, Newton's method is used to do this job in this work.

Scheme 3 (Runge-Kutta of Order 4)
We can also solve the first order differential system (7) The resulting system is of fourth order accuracy in both directions time and space, it is conditionally stable. In each time step, we need to solve four linear penta diagonal systems to find 1 3 , ,

Accuracy of the Proposed Schemes
To study the accuracy of Scheme 1, we replace the numerical solution n i U by the the exact solution n i u in (8) Taylor's expansion for all terms in (17) about grid point ( ) , m n x t , the following expressions are obtained 2  2  2  2 2  2  3 2  2  4  4  5 2   3  3  5   3  3  3  3  3  7  3  2  2  4  8  240 n n (19) By substituting these expressions into (17), and by collecting similar terms, we will get the local truncation error (LTE) as The first four brackets are zero by the KdV equation, LTE will be reduced to 240

Stability Analysis
In this section, we want study the stability of the proposed schemes [20]. Our  It is easy to see from (28) that, e 1 k α = , thus we can say that scheme 1 and scheme 2 are unconditionally stable according von-Neumann stability analysis.
For scheme 3 the numerical results showed that the scheme is conditionally stable (since it is explicit).

Numerical Results
In this part, several numerical examples are carefully designed to validate the efficiency of the proposed methods. In the physical opinion, the motion and interactions of solitons will be considered. In addition, some conservation laws that KdV equation satisfies will be examined by numerically calculating (using trapezoidal rule) the following three invariants corresponding to conservation of mass, momentum and energy as defined [7] ( )

Single Soliton
To test our numerical methods, we choose the initial condition ( ) ( ) subject to the homogenous boundary conditions. The exact solution for this test In order to generate the numerical solutions, the following parameters are In Table 1 and Table 2, we display the conserved quantities and the error for Scheme 1 and Scheme 2, respectively. The results show that the methods conserve the conserved quantities, with high accuracy. The simulation of the single soliton is given in Figure 1.
In Table 3, we present the results for a single solution using Scheme 3. These results are obtained by using the same previous parameters except 0 0.0001, 20 k x = = . From the numerical results, we have noticed that the method is conserves the conserved quantities exactly with high accuracy. The motion of single soliton displayed in Figure 2.

Rate of Convergence
To calculate the order of the proposed numerical schemes. We define the rate of convergence (RTC) as ( ) ( ) ( )

Two Solitons Interaction
As a second problem, we have discussed the behavior of the interaction of two solitary waves for Scheme (1) and (2) for different analytical solutions.

Sum of Two Single Solitons (a)
We choose initial condition a sum of two well separated single solitons:  Table   6 and Table 7. For the interaction scenario, see Figure 3.

Exact Two Solitons Solution (b)
In this test we pick our initial condition from the exact two soliton solution [2] [3] [21] ( ) ( ) ( )    Table 8 and Ta-  Figure 4 for 10 20 t − ≤ ≤ . We have noticed that, the two waves collide each other and leave the interaction region without any disturbance in their identities.

Two Solitons Solution (c)
We consider also the interaction of two solitary waves using the initial condition [3] ( ) ( By using (40) and the set of parameters 10 10 x − ≤ ≤ . The errors and conserved quantities are displayed in Table 10 and   Table 11. The results are highly accurate and the conserved quantities are almost constants. The interaction scenarios are displayed in Figure 5. We have noticed that the two waves interact and emerge after the interaction without any disturbance in their identities.
Using Scheme 3 with the initial condition (38) and the set of parameters To check the interaction of two solitons, in Table 12, we display the con-

Three Solitons Interaction
In this test, we want to study the interaction of three solitons having different amplitudes and traveling in the same direction. We choose the initial condition as sum of three well separated solitons of the form In Tables 13-15, we presented the conserved quantities for the proposed schemes. All methods showed the conservation of the conserved quantities. In Figure 7 and Figure 8, we display the interaction scenario of the three solitons. We have noticed that the three solitons recover their shapes after the interaction

Single Soliton Using Periodic Boundary Conditions
The final test in this work is to study the behavior of single soliton with periodic boundary conditions (for long time simulation) using Scheme 1. We select the set of parameters In Table 16, we presented the error and conserved quantities. We have noticed a high accuracy and exact conservation of the conserved quantities. In Figure 9, we display the motion of the single soliton for 0, ,50 t =  .

Conclusion
The numerical schemes for the KdV equation presented using Pade approximation of fourth order accuracy in space direction and second order accuracy in time using trapezoidal and implicit middle point rule. The resulting system is a