Volume 13, Issue 1
The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 253-280.

Published online: 2019-12

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• Abstract

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence  of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.

65M06, 65M12, 65M15

goldleaf0811@sina.com (Jinye Shen)

seuMathWxp@139.com (Xuping Wang)

zzsun@seu.edu.cn (Zhi-zhong Sun)

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@Article{NMTMA-13-253, author = {Shen , JinyeWang , Xuping and Sun , Zhi-zhong}, title = {The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {13}, number = {1}, pages = {253--280}, abstract = {

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence  of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0038}, url = {http://global-sci.org/intro/article_detail/nmtma/13439.html} }
TY - JOUR T1 - The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions AU - Shen , Jinye AU - Wang , Xuping AU - Sun , Zhi-zhong JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 253 EP - 280 PY - 2019 DA - 2019/12 SN - 13 DO - http://doi.org/10.4208/nmtma.OA-2019-0038 UR - https://global-sci.org/intro/article_detail/nmtma/13439.html KW - Nonlinear Korteweg-de Veries equation, difference scheme, existence, conservation, boundedness, convergence. AB -

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence  of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.

Jinye Shen, Xuping Wang & Zhi-zhong Sun. (2019). The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions. Numerical Mathematics: Theory, Methods and Applications. 13 (1). 253-280. doi:10.4208/nmtma.OA-2019-0038
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