A Compact-Type CIP Method for General Korteweg-de Vries Equation

and Applied Analysis 3 For the third derivatives at interior nodes, the following formula is given in [23]: α (u 󸀠󸀠󸀠 i−1 + u 󸀠󸀠󸀠 i+1 ) + u 󸀠󸀠󸀠 i = b u i+3 − 3u i+1 + 3u i−1 − u i−3


Introduction
The Korteweg-de Vries (KdV) equation, developed by Korteweg and de Vries [1] in 1895 to model weakly nonlinear waves, has been used in many different fields to model various physical phenomena of interest.In recent years, a number of numerical methods are proposed for solving KdV equations.In [2], Zabusky and Kruskal proposed a finite difference method for KdV equations.A local discontinuous Galerkin method was developed for solving KdV type equations containing third derivative terms in one and two space dimensions in [3].Numerical solutions for general KdV equation with Crank Nicolson method and B-spline FEM were compared with those obtained with Adomian decomposition method (ADM) in [4,5].Based on the multisymplectic theory, lots of numerical schemes were proposed for KdV equations [6][7][8].Ascher and McLachlan developed a simplified 8-point box scheme [9].By adding an artificial numerical condition to the periodic boundary, Wang et al. derived some new schemes and proved that they were more efficient than the Preissman scheme in [10].A completely explicit 6-point multisymplectic scheme is derived in [11].Recently many other numerical methods have been proposed for solving KdV type equations in [12][13][14][15][16][17][18].
In recent years, the less diffusive CIP scheme developed by Takewaki et al. [19] for solving hyperbolic equation has become very popular.However, the original CIP method [19][20][21][22] utilizing both the point values and its spatial gradients needs auxiliary boundary conditions for the spatial gradient information.Usually, it has to differentiate the equation with spatial variable to get the values of derivation on the node.For the simple case, where the velocity is constant, the procedure is not difficult, but it is not easy for complex equations.In 1992, Lele [23] developed high-order compact (HOC) difference schemes based on implicit interpolations for first and second derivatives.The implicit schemes are very accurate in smooth regions and have spectral-like resolution properties by using the global grid.High-order compact finite difference schemes coupled with high-order low-pass filter are applied to simulate KdV equations in [24].
In this paper, a new numerical scheme based on classical CIP and HOC schemes is proposed to solve KdV type equations.The new scheme is based on CIP scheme; as a new ingredient, the classical high-order compact scheme [23] is employed to obtain the derivatives rather than differentiate the equation with spatial variable to construct CIP scheme.By comparing with classical compact scheme for solving KdV equations, no filter is used to formulate the present scheme.
The paper is organized as follows.In Section 2, we give a brief description of CIP and high-order compact schemes.The numerical arithmetic of the present scheme is also discussed in the last part of this section.The implementation

Descriptions of Methods
In this paper, we consider the following generalized KdV equation [25]: The equation can be split into two parts: where  are real constants.We only consider the advective phase (2) to review CIP method.
2.1.The CIP Method.The CIP method in [21] uses cubicpolynomial interpolation to get the values of a function on nodes.The time evolution of spatial derivation is also required (4).We differentiate the advective phase of (2) with the spatial variable, and then the equation for derivatives of  can be obtained [26].Consider where  = / stands for the spatial derivatives of .The computational domain [, ] can be divided into  cells, and the cells are denoted by   = [  ,  +1 ].We only consider a uniform grid and the size of the cell by Δ = ( − )/.The cubic polynomial and the first-order derivative at the th step can be written as where  =  −   , and coefficients   ,   ,   , and   will be obtained with the following constrains: where  =  − sgn((  )), the sign sgn((  )) stands for the sign of (  ).Then, the coefficients of the cubic polynomial are given as follows: where Δ  =   −   .Thus, the values of  and  at the ( + 1)th step can be obtained as follows: We define   = −(  )Δ, and then the formulas are rewritten as It can be seen that only two points are used in CIP schemes to get  +1  .Then, the advantages of this method can be shown while the computational boundary is complex since less boundary points need to be handled.The CIP method uses only two neighboring stencils but maintains third-order accuracy.In this sense, high order accuracy is obtained though less computational stencils are used.More details for advantages of the CIP schemes can be found in [26].

High-Order Compact Scheme.
A series finite difference scheme to evaluate the spatial derivatives is presented in [23].The finite difference approximation to the derivative of the function is expressed as a linear combination of the given function values, and then the derivatives of the function are obtained by solving a tridiagonal or pentadiagonal system.Formulas for the first-order and the third-order derivatives are reviewed as below.More results for the approximation to derivatives can be found in [23,27].

The Derivatives at Interior Nodes.
In this paper, we consider the KdV equation on a uniform mesh, the spatial variable at the nodes is   =  × ℎ for 0 ≤  ≤  and the functions and the derivatives are denoted by   ,    .The firstorder derivatives for interior nodes are derived by writing approximations of the form [23]. Consider If the schemes are restricted to  ≥ 0 and  = 0, a oneparameter -family of fourth-order tridiagonal scheme is obtained.Consider A simple sixth-order tridiagonal scheme for first-order derivatives is given with  = 1/3,  = 0,  = 0,  = 14/9, and  = 1/9: For the third derivatives at interior nodes, the following formula is given in [23]: The fourth-order tridiagonal schemes can be obtained with the coefficients  = 2 and  = 2−1.The compact tridiagonal scheme is given with  = 1/2,  = 2, and  = 0.And the sixthorder tridiagonal scheme is given with  = 7/16,  = 2, and  = −1/8.

Nonperiodic Boundaries.
For those near boundary nodes, approximation formulas for the first-order derivatives of nonperiodic boundary problems are given by one-side formulation as follows [23]: The coefficients for schemes of the third and fourth order are given by Third order: Fourth order: The sixth-order scheme also be given, for those near boundary nodes, six order approximation formulas for the firstorder derivatives can be written as follows: where For the second point, the formula is where The dissymmetry condition is used for the th and ( − 1)th points.
2.3.The Present Compact-Type CIP Method.In this section, a new compact-type CIP scheme is proposed for (1).For simplicity, the following equation is considered to introduce the method.Consider where  and  are constants.We split the solution of equation into two phases: We consider a 1D mesh, consisting of ( + 1) points:  0 ,  1 ,  2 , . ..,  −1 ,   ; the values of  on these nodes at the th step are denoted by   0 ,   1 . ..,   −1 ,    .At first, ( 21) is considered and CIP method is applied to the equation.The cubic polynomial at the th time stage is where  =   −  and the coefficient    ,    , and    are given by (7), where () =  and    denote the derivative of    at the th node.The predictor-corrector scheme is employed to calculate the value  * .
In the present method, the values    , 0 ≤  ≤  are expressed as a linear combination of the given values    , 0 ≤  ≤ .On the other hand, the HOC method is employed to evaluate the derivatives    , 0 ≤  ≤ .A simple sixthorder tridiagonal scheme for interior points and boundary points is used in this paper.
Temporal discretization for ( 22) can be solved by using third-order Runge-Kutta method: The HOC scheme ( 13) is used to solve the third derivatives   in (24).The sixth-order tridiagonal scheme with the periodic boundary condition is used in this paper.
The essential ingredients of the computational algorithm for (20)   (1) CIP method is used to obtain  * .
(a) The values of the first derivative on the all nodes are obtained by using the HOC scheme (12).(b) Predictor-corrector CIP scheme is as follows: (i) predictor step: where   = −   Δ.We also get  * * * at the ( + 1/2)th time stage by using linear interpolation or QUICK scheme based on the value    ; (ii) corrector step (CIP method): where  ⬦ = (1/2)( * * +  * * * ); (iii) the predictor and corrector step are employed again to get  * .
(a) The HOC scheme ( 13) is used to obtain thirdorder derivatives.(b) Temporal discretization for ( 22) can be solved by using third-order Runge-Kutta method.
The predictor-corrector scheme is an important step in the present method.Periodic boundary condition is applied to (22).

Numerical Results
In this section, some numerical tests for KdV and general KdV equations are carried out.The discrete  2 and  ∞ error norms are defined as follows: where  and ũ are exact and numerical solution, respectively.For KdV equations, there are an infinite number of conservation laws [28].We will focus our analysis on the following three conservation laws: where  1 ,  2 , and  3 represent mass, momentum, and energy.The nonperiodic boundary formulation is applied to (21) (HOC approximation formulas for the third-order derivatives are used) and periodic boundary conditions for thirdorder derivatives are used in the following examples.
Example 1.In this example, we consider the following classical KdV equation: The analytical solution for (29) is The time-steps are set by the relation Δ = (Δ) 3 .The  2 and  ∞ errors, orders, invariants, and time costs at time  = 0.1 are illustrated in Table 2.It can be observed that the proposed scheme is third-order accurate in the spatial dimension.It is well known that high-order TVD Runge-Kutta methods suffer from small time-step restrictions.In this case, we observe that numerical errors are still not dominated by the spatial discretization with the relation Δ = (Δ) 3 .
Table 1 indicates  2 and  ∞ errors and invariants with  = 100 at time  = 0.1, 0.2, . . ., 0.9.The present method can also be shown to have the conservative property.
Example 2. In this example, we consider the following classical KdV equation: The numerical solutions are obtained with  = 0.2 and  = 0.0001; numerical and exact solutions at times  = 1, 3, 5 are presented in Figure 1; the figure shows that the numerical dissipation of soliton is very small.
Example 3. The general KdV equation is presented by the following equation [29]: Consider the initial value problem associated with (33) using the initial condition for  = 2: where  = 4.The numerical solutions for  = 2 are obtained with  = 0.1 and  = 0.05 ×  3 for  = 2.The progress of the numerical and analytical solutions at times  = 1, 2 and the absolute error at  = 1 are shown in Figure 2. The  ∞ and  2 error estimates for the case  = 2 are given in Table 3.
For the case  = 1 with the initial condition [14] and the analytical solution is The  ∞ and  2 error estimates with  = 0.2 are given in Table 4 for  = 1, 2, 3, 4, 5, from which it is not difficult to see that the present results are comparable with those present in [14].The numerical and the analytical solutions at times  = 1, 3, 5 and the absolute error for  = 3 are shown in Figure 3.

Figure 1 :
Figure 1: Numerical and analytical solutions for equation   +   +   = 0 of Example 2 at various time stages.

Figure 2 :
Figure 2: Numerical and analytical solutions (line) for equation   + 6 2   +   = 0 of Example 3 at various time stages and the absolute error for  = 1.
are presented below.Suppose we have got the values    .The values  +1  are given as follows.

Table 2 :
Numerical errors and invariants with  = 100 at different times.