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Numerical solution of Kortewegde VriesBurgers equation by the compacttype CIP method
Advances in Difference Equations volumeÂ 2015, ArticleÂ number:Â 353 (2015)
Abstract
In this paper, a hybrid compactCIP scheme is proposed to solve Kortewegde VriesBurgers equation. The nonlinear advective terms are computed based on the classical constrained interpolation profile (CIP) method, which is coupled with a highorder compact scheme for thirdorder derivatives in Kortewegde VriesBurgers equation. The strong stability preserving thirdorder RungeKutta time discretizations is adopted in this work. A test case is presented to demonstrate the highresolution properties of the proposed compactCIP scheme.
1 Introduction
In 1895, Korteweg and de Vries [1] developed the Korteweg de Vries (KdV) equation to model weakly nonlinear waves. It has been used in several different fields to describe various physical phenomena of interest. The KdVBurgers (KdVB) equation which is derived by Su and Gardner [2] appears in the study of the weak effects of dispersion, dissipation, and nonlinearity in waves propagating in a liquidfilled elastic tube. Recently, the nonlinear fractional partial differential equations, such as fractional KdVBurgers equation [3], fractional SchrÃ¶dingerKortewegde Vries equations [4] and fractional Burgersâ€™ equations [5], were also presented to describe many important phenomena and dynamic processes in physics. Some theoretical issues concerning the KdVB equation, such as the traveling wave solution, have received considerable attention [6]. A number of exact solitary wave solutions to KdVB equations have been found in the past few years. The exact solutions of a compound KdVB equation were obtained by using a homogeneous balance method in [7]. By using the special truncated expansion method, Hassan [8] constructed solitary wave solutions for the compound KdVB equation and discussed the generalized twodimensional KdVB equation. The Expfunction method is applied to obtain generalized solitary solutions and periodic solutions for the KdVB equation in [9]. In the past several decades, many authors have paid attention to studying the numerical methods for solving KdVB equations. Soliman extended the variational iterations method to solve the KdVB equations [10]. A new decomposition method was presented to find the explicit and numerical solutions of the KdVB equations without any transformations, linearization or weak nonlinearity assumptions in [11]. The elementfree Galerkin (EFG) method for numerically solving the compound KdVB equation was discussed by RongJun and YuMin in [12]. The explicit restrictive Taylor approximation (RTA) was implemented to find numerical solution of KdVBurgers in [13]. Nonlinear dispersive wave propagation problems that described the KdVB equations in [14] were simulated by highorder compact finite difference schemes coupled with highorder lowpass filter and the classical fourthorder RungeKutta scheme.
In 1992, based on implicit interpolations, highorder compact (HOC) difference schemes for different derivatives were developed by Lele [15]. These implicit schemes were very accurate in smooth regions, and they have spectrallike resolution properties by using the global grid. Li and Visbal applied the compact schemes coupled with highorder lowpass filter for solving KdVBurgers equations in [14]. In the past few years, it has been popular for using the less diffusive and less oscillating CIP scheme which was developed by Takewaki et al. [16] to solve hyperbolic equation. The classical CIP schemes which were essentially written as the semiLagrangian formulation were lowdiffusion and stable. The scheme can solve hyperbolic equations with thirdorder accuracy in space [17]. However, the original CIP method [16, 18â€“20] utilizes auxiliary boundary conditions for the spatial gradient information. Usually, in order to get the values of derivation on the node, it has to differentiate the equation with spatial variable. The procedure is easy while the velocity is constant, but it is difficult for complex equations. By using the compact scheme for the values of derivation on the nodes, we present a new compact scheme based on the characteristic method for solving KdVBurgers equations.
In this paper, a new numerical method named compacttype CIP schemes based on combination of CIP and highorder compact schemes is advanced to solve the KdVBurgers equations. The present scheme is mainly based on the idea of characteristic method; as a new ingredient, the highorder compact scheme is employed to obtain the derivatives rather than differentiate the equation with spatial variable to construct a CIP scheme, and then resolution properties can also be obtained. By comparing with the classical compact scheme for solving KdVBurgers equations, no filter is used to overcome nonphysical oscillations.
The remainder of the paper is organized as follows. In SectionÂ 2, CIP is described in brief, then highorder compact schemes are given. The numerical algorithm of the present schemes is described in this section. The merit of our present method for KdVB equation is displayed in SectionÂ 3, a comparison of numerical solutions with exact solutions is carried out to illustrate the capability of the method for nonlinear dispersive equations. At last, aÂ short discussion of the present method is given in SectionÂ 4.
2 Descriptions of methods
In this paper, we consider the following generalized KdVBurgers equation [8]:
where Î±, Î², Î³ and Î´ are real constants. The equation can be split into two parts
where \(a(u)=(\alpha+\beta u)u\).
2.1 The CIP method
In this section, we review the CIP method briefly. The CIP method in [19] uses cubicpolynomial interpolation to get the values of function on nodes. The primary goal of the numerical algorithm will be to retrieve the lost information inside the grid cell between these digitized points. We differentiate the advective phase of equation (2.2) with the spatial variable x, then we get [21]
where \(g=\partial u/\partial x\) stands for the spatial derivatives of u. For the computational domain \([a,b]\), we only consider a uniform grid with a space step \(\Delta x=\frac{ba}{N}\). If both the values of u and g are given at two grid points, the cubic polynomial at the nth step can be written as
where \(X=xx_{i}\), and coefficients \(a_{i}\), \(b_{i}\), \(c_{i}\) and \(d_{i}\) will be obtained with the following constrains:
where \(iup=i\operatorname{sgn}(a(u_{i}))\), the sign \(\operatorname{sgn}(a(u_{i}))\) stands for the sign of \(a(u_{i})\). Then the coefficients of the cubic polynomial are given
where \(\Delta x_{i}=x_{iup}x_{i}\). Thus, the profile u and g at the \((n+1)\)th step can be obtained by shifting the profile by \(a(u_{i})\Delta t\)
We define \(\xi_{i}=a(u_{i})\Delta t\), then the formulates are rewritten as
It can be seen that we only use two points in CIP schemes to get \(u_{i}^{n+1}\). Then we display the implementation of this method, while the computational boundary is complex and less boundary points need to be handled. The CIP method uses only two neighboring stencils, but keeps thirdorder precision. In this sense, highorder precision is gained though less computational stencils are used. For more details about CIP schemes, readers can refer to [21].
2.2 Highorder compact scheme
Lele developed highorder linear compact difference schemes based on implicit interpolations in [15]. These implicit schemes are very accurate in smooth regions and have spectrallike resolution properties by using the global grid. The finite difference approximation to the derivative of the function is expressed as a linear combination of the given function values, then, by solving a tridiagonal or pentadiagonal system, the derivatives of the function can be obtained. In this section, a review of formulas for firstorder, secondorder and thirdorder derivatives is presented. For more details about the highorder compact schemes, readers can refer to [15, 22].
2.2.1 The derivatives at interior nodes
In this paper, the KdVB equation on a uniform mesh is considered, the point values and the derivatives are indicated by \(u_{i}\), \(u_{i}'\), \(i=1,\ldots, N\). For the firstorder derivatives at interior nodes, we have the formula [15]
If the schemes are restricted to \(\beta\geq0\) and \(c=0\), this provides a oneparameter Î±family of fourthorder tridiagonal scheme with
A simple sixthorder tridiagonal scheme is given by the coefficients
The scheme can be rewritten as follows:
For the secondorder derivatives at interior nodes, we have the formula [15]
which provides a oneparameter Î±family of fourthorder tridiagonal schemes with
A sixthorder tridiagonal scheme is also given with
then the sixthorder tridiagonal scheme for (2.14) can be rewritten as follows
For the thirdorder derivatives at interior nodes, the following formula is given in [15]:
which provides an Î±family of fourthorder tridiagonal schemes with \(a=2\), \(b=2\alpha1\). The simple sixthorder tridiagonal scheme is given with \(\alpha=\frac{7}{16}\), \(a=2\), \(b=\frac{1}{8}\).
2.2.2 Nonperiodic boundaries
For those near boundary nodes, approximation formulas for the firstorder derivatives of nonperiodic boundary problems are given by oneside formulation as follows [15]:
The coefficients for the schemes of third and fourthorder derivatives are given by
The sixthorder scheme is also given, where the first and second points need to be handled. For the first point, the formula is
where
For the second point, the formula is
where
The dissymmetry condition is used for the Nth and \((N1)\)th points.
The boundary formulations for the secondorder derivatives also were constructed in [15].
For the thirdorder accuracy, the coefficients are given as follows:
2.3 The proposed compacttype CIP method
In this section, a new compacttype CIP scheme is proposed for equation (2.1). The present scheme is mainly based on the idea of characteristic method; as a new ingredient, the highorder compact scheme is employed to obtain the derivatives rather than differentiate the equation with spatial variable to construct a CIP scheme. To explain the present scheme, we consider the KdVB equations as follows:
where Î± and Î´ are constants. We split the solution of equation into two phases
We consider a 1D grid with \(x_{0}, x_{1}, x_{2},\ldots, x_{N1},x_{N}\). At the nth step, the point values of u are denoted by \(u_{0}^{n}, u_{1}^{n} ,\ldots, u_{N1}^{n},u_{N}^{n}\). At first, CIP method is applied to the advective equation (2.28). If both the values of \(u_{i}\) and \(u_{i}^{\prime n}\) are given at two grid points, the cubic polynomial at the nth time stage can be written as follows:
where \(X=x_{i}x\), the coefficients \(a_{i}^{n}\), \(b_{i}^{n}\), \(c_{i}^{n}\) are given by (2.7). The predictorcorrector scheme is employed to calculate the value \(u^{*}\).
To formulate the classical CIP scheme, equation (2.4) was used to get the values of \(u_{i}^{\prime n}\). In the present method, the highorder compact scheme (2.10) is employed to evaluate the derivatives \(u_{i}^{\prime n}\), \(0\leq i\leq N\). In this paper, we use a simple sixthorder tridiagonal scheme for interior points and boundary points, then the coefficients \(a_{i}^{n}\), \(b_{i}^{n}\), \(c_{i}^{n}\) in (2.7) can be obtained. Temporal discretization for equation (2.29) can be solved by using a thirdorder RungeKutta method as follows:
where \(L(u)=\gamma u_{xx}+\delta u_{xxx}\). The highorder compact formulas (2.17) and (2.18) are used to solve the second and thirdorder derivatives in equation (2.31). In this paper, we use the sixthorder tridiagonal scheme with the periodic boundary condition.
Supposing the values \(u_{i}^{n}\) have been obtained, the essential ingredients of the computational algorithm for equation (2.27) consist of the following steps:

1.
CIP method is used to obtain \(u^{*}\)

a.
The values of the firstorder derivative on all the nodes are obtained by using the HOC scheme (2.13).

b.
Predictorcorrector CIP scheme:

(a)
Predictor step
$$u_{i}^{**}=U_{i}^{n} \bigl(x_{i}\alpha u_{i}^{n}\Delta t \bigr)=a_{i}^{n}{\xi_{i}}^{3}+b_{i}^{n}{ \xi_{i}}^{2}+c_{i}^{n} \xi_{i}+u_{i}^{n}, $$where \(\xi_{i}=\alpha u_{i}^{n}\Delta t\). We also get \(u^{***}\) at the (n+\(\frac{1}{2}\))th time stage by using linear interpolation or QUICK scheme based on the value \(u^{n}_{i}\).

(b)
Corrector step (CIP method)
$$\hat{u}_{i}^{*}=U_{i}^{n} \bigl(x_{i}\alpha u_{i}^{\diamond}\Delta t \bigr)=a_{i}^{n}{\xi_{i}}^{3}+b_{i}^{n}{ \xi_{i}}^{2}+c_{i}^{n} \xi_{i}+u_{i}^{n}, $$where \(u^{\diamond}=\frac{1}{2}(u^{**}+u^{***})\).

(c)
The predictor and corrector steps are employed again to get \(u^{*}\).

(a)

a.

2.
Highorder compact schemes and RungeKutta method for solving equation (2.29)
3 Numerical results
In this section, we provide a numerical example with two different initial conditions for the present compactCIP scheme with the thirdorder SSP RungeKutta time discretization. The nonperiodic boundary formulation is applied to (2.28) (HOC approximation formulas for first and secondorder derivatives are used) and periodic boundary conditions for thirdorder derivatives in the following example.
Example 3.1
We consider the KdVB equation
with the initial solution for \(\gamma=0\), \(\delta=1\)
We show the numerical solutions for different values of Î± and Î² in FigureÂ 1. If we let \(\beta=0\), \(\alpha=2\), \(\gamma=5\), \(\delta=3\). The exact solution for this case is [14]
where \(\theta=\frac{1}{3}x+\frac{2}{3}t\). The numerical and analytical solutions are shown in FigureÂ 2. The numerical solutions are identical to the exact solution.
4 Conclusions
In this paper, a highorder compactCIP scheme is applied to simulate Kortewegde Vries Burgers equations. The proposed scheme is mainly based on the idea of characteristic method; as a new ingredient, the highorder compact scheme is employed to obtain the derivatives rather than differentiate the equation with spatial variable to construct a CIP scheme, and then resolution properties can also be obtained. The numerical results show the good performance and high resolution property of the proposed scheme.
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Acknowledgements
The work is partly supported by the Fundamental Research Funds for the Central Universities (2012QNB07, 2015QNA46) and Universities Provincial Natural Science Research Project of Anhui Province (KJ2014B17).
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Shi, Y., Xu, B. & Guo, Y. Numerical solution of Kortewegde VriesBurgers equation by the compacttype CIP method. Adv Differ Equ 2015, 353 (2015). https://doi.org/10.1186/s1366201506825
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DOI: https://doi.org/10.1186/s1366201506825