Volume 13, Issue 5
A Note on the Convergence of a Crank-Nicolson Scheme for the KdV Equation

Int. J. Numer. Anal. Mod., 13 (2016), pp. 657-675.

Published online: 2016-09

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• Abstract

The aim of this paper is to establish the convergence of a fully discrete Crank-Nicolson type Galerkin scheme for the Cauchy problem associated to the KdV equation. The convergence is achieved for initial data in $L^2$, and we show that the scheme converges strongly in $L^2(0, T; L^2_{loc}(\mathbb{R}))$ to a weak solution for some $T >0$. Finally, the convergence is illustrated by a numerical example.

• Keywords

Crank-Nicolson scheme, KdV equation.

65M99, 70H06, 74J35

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@Article{IJNAM-13-657, author = {}, title = {A Note on the Convergence of a Crank-Nicolson Scheme for the KdV Equation}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {5}, pages = {657--675}, abstract = {

The aim of this paper is to establish the convergence of a fully discrete Crank-Nicolson type Galerkin scheme for the Cauchy problem associated to the KdV equation. The convergence is achieved for initial data in $L^2$, and we show that the scheme converges strongly in $L^2(0, T; L^2_{loc}(\mathbb{R}))$ to a weak solution for some $T >0$. Finally, the convergence is illustrated by a numerical example.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/458.html} }
TY - JOUR T1 - A Note on the Convergence of a Crank-Nicolson Scheme for the KdV Equation JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 657 EP - 675 PY - 2016 DA - 2016/09 SN - 13 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/458.html KW - Crank-Nicolson scheme, KdV equation. AB -

The aim of this paper is to establish the convergence of a fully discrete Crank-Nicolson type Galerkin scheme for the Cauchy problem associated to the KdV equation. The convergence is achieved for initial data in $L^2$, and we show that the scheme converges strongly in $L^2(0, T; L^2_{loc}(\mathbb{R}))$ to a weak solution for some $T >0$. Finally, the convergence is illustrated by a numerical example.

R. Dutta & N. H. Risebro. (1970). A Note on the Convergence of a Crank-Nicolson Scheme for the KdV Equation. International Journal of Numerical Analysis and Modeling. 13 (5). 657-675. doi:
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