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- Open Access
Numerical solution of Korteweg-de Vries-Burgers equation by the compact-type CIP method
- YuFeng Shi^{1},
- Biao Xu^{2, 3}Email author and
- Yan Guo^{4}
https://doi.org/10.1186/s13662-015-0682-5
© Shi et al. 2015
Received: 10 September 2015
Accepted: 29 October 2015
Published: 18 November 2015
Abstract
In this paper, a hybrid compact-CIP scheme is proposed to solve Korteweg-de Vries-Burgers equation. The nonlinear advective terms are computed based on the classical constrained interpolation profile (CIP) method, which is coupled with a high-order compact scheme for third-order derivatives in Korteweg-de Vries-Burgers equation. The strong stability preserving third-order Runge-Kutta time discretizations is adopted in this work. A test case is presented to demonstrate the high-resolution properties of the proposed compact-CIP scheme.
Keywords
- high-order compact schemes
- CIP schemes
- Korteweg-de Vries-Burgers equation
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 KdV-Burgers (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 liquid-filled elastic tube. Recently, the nonlinear fractional partial differential equations, such as fractional KdV-Burgers equation [3], fractional Schrödinger-Korteweg-de Vries equations [4] and fractional Burgers’ 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 two-dimensional KdVB equation. The Exp-function 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 element-free Galerkin (EFG) method for numerically solving the compound KdVB equation was discussed by Rong-Jun and Yu-Min in [12]. The explicit restrictive Taylor approximation (RTA) was implemented to find numerical solution of KdV-Burgers in [13]. Nonlinear dispersive wave propagation problems that described the KdVB equations in [14] were simulated by high-order compact finite difference schemes coupled with high-order low-pass filter and the classical fourth-order Runge-Kutta scheme.
In 1992, based on implicit interpolations, high-order compact (HOC) difference schemes for different derivatives were developed by Lele [15]. These implicit schemes were very accurate in smooth regions, and they have spectral-like resolution properties by using the global grid. Li and Visbal applied the compact schemes coupled with high-order low-pass filter for solving KdV-Burgers 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 semi-Lagrangian formulation were low-diffusion and stable. The scheme can solve hyperbolic equations with third-order accuracy in space [17]. However, the original CIP method [16, 18–20] 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 KdV-Burgers equations.
In this paper, a new numerical method named compact-type CIP schemes based on combination of CIP and high-order compact schemes is advanced to solve the KdV-Burgers equations. The present scheme is mainly based on the idea of characteristic method; as a new ingredient, the high-order 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 KdV-Burgers equations, no filter is used to overcome non-physical oscillations.
The remainder of the paper is organized as follows. In Section 2, CIP is described in brief, then high-order 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 Section 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, a short discussion of the present method is given in Section 4.
2 Descriptions of methods
2.1 The CIP method
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 third-order precision. In this sense, high-order precision is gained though less computational stencils are used. For more details about CIP schemes, readers can refer to [21].
2.2 High-order compact scheme
Lele developed high-order linear compact difference schemes based on implicit interpolations in [15]. These implicit schemes are very accurate in smooth regions and have spectral-like 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 first-order, second-order and third-order derivatives is presented. For more details about the high-order compact schemes, readers can refer to [15, 22].
2.2.1 The derivatives at interior nodes
2.2.2 Non-periodic boundaries
2.3 The proposed compact-type CIP method
- 1.CIP method is used to obtain \(u^{*}\)
- a.
The values of the first-order derivative on all the nodes are obtained by using the HOC scheme (2.13).
- b.Predictor-corrector CIP scheme:
- (a)Predictor stepwhere \(\xi_{i}=-\alpha u_{i}^{n}\Delta t\). We also get \(u^{***}\) at the (n+\(\frac{1}{2}\))th time stage by using linear interpolation or QUICK scheme based on the value \(u^{n}_{i}\).$$u_{i}^{**}=U_{i}^{n} \bigl(x_{i}-\alpha u_{i}^{n}\Delta t \bigr)=a_{i}^{n}{\xi_{i}}^{3}+b_{i}^{n}{ \xi_{i}}^{2}+c_{i}^{n} \xi_{i}+u_{i}^{n}, $$
- (b)Corrector step (CIP method)where \(u^{\diamond}=\frac{1}{2}(u^{**}+u^{***})\).$$\hat{u}_{i}^{*}=U_{i}^{n} \bigl(x_{i}-\alpha u_{i}^{\diamond}\Delta t \bigr)=a_{i}^{n}{\xi_{i}}^{3}+b_{i}^{n}{ \xi_{i}}^{2}+c_{i}^{n} \xi_{i}+u_{i}^{n}, $$
- (c)
The predictor and corrector steps are employed again to get \(u^{*}\).
- (a)
- a.
- 2.
3 Numerical results
In this section, we provide a numerical example with two different initial conditions for the present compact-CIP scheme with the third-order SSP Runge-Kutta time discretization. The non-periodic boundary formulation is applied to (2.28) (HOC approximation formulas for first- and second-order derivatives are used) and periodic boundary conditions for third-order derivatives in the following example.
Example 3.1
4 Conclusions
In this paper, a high-order compact-CIP scheme is applied to simulate Korteweg-de Vries Burgers equations. The proposed scheme is mainly based on the idea of characteristic method; as a new ingredient, the high-order 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.
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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