[1] Wang Xuping, Sun Zhizhong,. A second-order convergent and linearized difference schemefor the initial-boundary value problemof the Korteweg-de Vries equation [J]. Journal of Southeast University (English Edition), 2022, 38 (2): 203-212. [doi:10.3969/j.issn.1003-7985.2022.02.013]

Copy

A second-order convergent and linearized difference schemefor the initial-boundary value problemof the Korteweg-de Vries equation()

Share：

Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

38

Issue:

2022 2

Page:

203-212

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2022-06-20

Info

Title:

A second-order convergent and linearized difference schemefor the initial-boundary value problemof the Korteweg-de Vries equation

Author(s):

Wang Xuping;  Sun Zhizhong

School of Mathematics, Southeast University, Nanjing 210096, China

Keywords:

Korteweg-de Vries(KdV)equation;  linearized difference scheme;  conservation;  convergence

PACS:

O241.82

DOI:

10.3969/j.issn.1003-7985.2022.02.013

Abstract:

To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation, an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order. Then, a difference scheme is constructed for the system. The new variable introduced can be separated from the difference scheme to obtain another difference scheme containing only the original variable. The energy method is applied to the theoretical analysis of the difference scheme. Results show that the difference scheme is uniquely solvable and satisfies the energy conservation law corresponding to the original problem. Moreover, the difference scheme converges when the step ratio satisfies a constraint condition, and the temporal and spatial convergence orders are both two. Numerical examples verify the convergence order and the invariant of the difference scheme. Furthermore, the step ratio constraint is unnecessary for the convergence of the difference scheme. Compared with a known two-level nonlinear difference scheme, the proposed difference scheme has more advantages in numerical calculation.

References:

[1] Korteweg D J, de Vries G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[J]. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1895, 39(240): 422-443. DOI:10.1080/14786449508620739.
[2] Kuo P Y, Wu H M. Numerical solution of K.D.V. equation[J]. Journal of Mathematical Analysis and Applications, 1981, 82(2): 334-345. DOI:10.1016/0022-247X(81)90199-2.
[3] Zhu S H. A scheme with a higher-order discrete invariant for the KdV equation[J]. Applied Mathematics Letters, 2001, 14(1): 17-20. DOI:10.1016/S0893-9659(00)00105-1.
[4] Taha T R, Ablowitz M J. Analytical and numerical aspects of certain nonlinear evolution equations. Ⅰ. Analytical[J]. Journal of Computational Physics, 1984, 55(2): 192-202. DOI:10.1016/0021-9991(84)90002-0.
[5] Shen J Y, Wang X P, Sun Z Z. The conservation and convergence of two finite difference schemes for KdV equations with initial and boundary value conditions[J]. Numerical Mathematics: Theory, Methods and Applications, 2020, 13(1): 253-280. DOI:10.4208/nmtma.oa-2019-0038.
[6] Wang X P, Sun Z Z. A second order convergent difference scheme for the initial-boundary value problem of Korteweg-de Vires equation[J]. Numerical Methods for Partial Differential Equations, 2021, 37(5): 2873-2894. DOI:10.1002/num.22646.
[7] Alexander M E, Morris J L.Galerkin methods applied to some model equations for non-linear dispersive waves[J]. Journal of Computational Physics, 1979, 30(3): 428-451. DOI:10.1016/0021-9991(79)90124-4.
[8] Yan J, Shu C W. A local discontinuous Galerkin method for KdV type equations[J]. SIAM Journal on Numerical Analysis, 2002, 40(2): 769-791. DOI:10.1137/s0036142901390378.
[9] Xu Y, Shu C W. Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations[J]. Computer Methods in Applied Mechanics and Engineering, 2007, 196(37/38/39/40): 3805-3822. DOI:10.1016/j.cma.2006.10.043.
[10] Shen J. A new dual-petrov-Galerkin method for third and higher odd-order differential equations: Application to the KDV equation[J]. SIAM Journal on Numerical Analysis, 2003, 41(5): 1595-1619. DOI:10.1137/s00361429-02410271.
[11] Feng B F, Wei G W. A comparison of the spectral and the discrete singular convolution schemes for the KdV-type equations[J]. Journal of Computational and Applied Mathematics, 2002, 145(1): 183-188. DOI:10.1016/S0377-0427(01)00543-X.
[12] Zhang Z Q, Ma H P. A rational spectral method for the KdV equation on the half line[J]. Journal of Computational and Applied Mathematics, 2009, 230(2): 614-625. DOI:10.1016/j.cam.2009.01.025.
[13] Dag, Dereli Y. Numerical solutions of KdV equation using radial basis functions[J]. Applied Mathematical Modelling, 2008, 32(4): 535-546. DOI:10.1016/j.apm.2007.02.001.
[14] Shen Q. A meshless method of lines for the numerical solution of KdV equation using radial basis functions[J]. Engineering Analysis With Boundary Elements, 2009, 33(10): 1171-1180. DOI:10.1016/j.enganabound.2009.04.008.
[15] Dehghan M, Shokri A. A numerical method for KdV equation using collocation and radial basis functions[J]. Nonlinear Dynamics, 2007, 50(1/2): 111-120. DOI:10.1007/s11071-006-9146-5.
[16] Bhatta D D, Bhatti M I. Numerical solution of KdV equation using modified Bernstein polynomials[J]. Applied Mathematics and Computation, 2006, 174(2): 1255-1268. DOI:10.1016/j.amc.2005.05.049.
[17] Zhang B Y. Boundary stabilization of the Korteweg-de Vries equation[C]//Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena, 1994: 371-389. DOI:10.1007/978-3-0348-8530-0_21.
[18] Sun H, Sun Z Z. On two linearized difference schemes for Burgers’ equation[J]. International Journal of Computer Mathematics, 2015, 92(6): 1160-1179. DOI:10.1080/00207160.2014.927059.
[19] Mohamad M N B. Exact solutions to the combined KdV and mKdV equation[J]. Mathematical Methods in the Applied Sciences, 1992, 15(2): 73-78. DOI:10.1002/mma.1670150202.

Memo

Memo:

Biographies: Wang Xuping(1995—), female, Ph.D. candidate; Sun Zhizhong(corresponding author), male, doctor, professor, zzsun@seu.edu.cn.
Foundation item: The National Natural Science Foundation of China(No. 11671081).
Citation: Wang Xuping, Sun Zhizhong. A second-order convergent and linearized difference scheme for the initial-boundary value problem of the Korteweg-de Vries equation[J].Journal of Southeast University(English Edition), 2022, 38(2):203-212.DOI:10.3969/j.issn.1003-7985.2022.02.013.

Common functions