Stability and Convergence analysis of a Crank–Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

Mukul Dwivedi; Tanmay Sarkar

doi:10.5802/smai-jcm.108

Mukul Dwivedi ¹; Tanmay Sarkar ¹

¹ Department of Mathematics, Indian Institute of Technology Jammu, Jagti, NH-44 Bypass Road, Post Office Nagrota, Jammu - 181221, India

The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 107-139.

Abstract

In this paper we study the convergence of a fully discrete Crank–Nicolson Galerkin scheme for the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation, which involves the fractional Laplacian and non-linear convection terms. Our proof relies on the Kato type local smoothing effect to estimate the localized $H^{α / 2}$ -norm of the approximated solution, where $α \in [1, 2)$ . We demonstrate that the scheme converges strongly in $L^{2} (0, T; L_{l o c}^{2} (ℝ))$ to a weak solution of the fractional KdV equation provided the initial data in $L^{2} (ℝ)$ . Assuming the initial data is sufficiently regular, we obtain the rate of convergence for the numerical scheme. Finally, the theoretical convergence rates are justified numerically through various numerical illustrations.

Published online: 2024-06-13

MR Zbl

DOI: 10.5802/smai-jcm.108

Classification: 35Q53, 65M60, 65M12
Keywords: Fractional Korteweg-de Vries equation, fractional Laplacian, Galerkin method, rate of convergence, $L^2$ initial data

Author's affiliations:

Mukul Dwivedi ¹; Tanmay Sarkar ¹

¹ Department of Mathematics, Indian Institute of Technology Jammu, Jagti, NH-44 Bypass Road, Post Office Nagrota, Jammu - 181221, India

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     title = {Stability and {Convergence} analysis of a {Crank{\textendash}Nicolson} {Galerkin} scheme for the fractional {Korteweg-de} {Vries} equation},
     journal = {The SMAI Journal of computational mathematics},
     pages = {107--139},
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Mukul Dwivedi; Tanmay Sarkar. Stability and Convergence analysis of a Crank–Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation. The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 107-139. doi : 10.5802/smai-jcm.108. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.108/

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