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 -norm of the approximated solution, where . We demonstrate that the scheme converges strongly in to a weak solution of the fractional KdV equation provided the initial data in . 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.
Keywords: Fractional Korteweg-de Vries equation, fractional Laplacian, Galerkin method, rate of convergence, $L^2$ initial data
Mukul Dwivedi 1; Tanmay Sarkar 1
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@article{SMAI-JCM_2024__10__107_0, author = {Mukul Dwivedi and Tanmay Sarkar}, 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}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {10}, year = {2024}, doi = {10.5802/smai-jcm.108}, language = {en}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.108/} }
TY - JOUR AU - Mukul Dwivedi AU - Tanmay Sarkar TI - Stability and Convergence analysis of a Crank–Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation JO - The SMAI Journal of computational mathematics PY - 2024 SP - 107 EP - 139 VL - 10 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.108/ DO - 10.5802/smai-jcm.108 LA - en ID - SMAI-JCM_2024__10__107_0 ER -
%0 Journal Article %A Mukul Dwivedi %A Tanmay Sarkar %T Stability and Convergence analysis of a Crank–Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation %J The SMAI Journal of computational mathematics %D 2024 %P 107-139 %V 10 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.108/ %R 10.5802/smai-jcm.108 %G en %F SMAI-JCM_2024__10__107_0
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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