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

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     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},
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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/

References
Cited by

[1] Laziz Abdelouhab; Jerry L. Bona; M. Felland; Jean-Claude Saut Nonlocal models for nonlinear, dispersive waves, Physica D, Volume 40 (1989) no. 3, pp. 360-392 | DOI | MR | Zbl

[2] Umberto Biccari; Mahamadi Warma; Enrique Zuazua Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., Volume 17 (2017) no. 2, pp. 387-409 | DOI | MR | Zbl

[3] Philippe G. Ciarlet The Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, 2002 | DOI

[4] Eleonora Di Nezza; Giampiero Palatucci; Enrico Valdinoci Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521-573 | DOI | MR | Zbl

[5] Rajib Dutta; Helge Holden; Ujjwal Koley; Nils Henrik Risebro Operator splitting for the Benjamin–Ono equation, J. Differ. Equations, Volume 259 (2015) no. 11, pp. 6694-6717 | DOI | MR | Zbl

[6] Rajib Dutta; Helge Holden; Ujjwal Koley; Nils Henrik Risebro Convergence of finite difference schemes for the Benjamin–Ono equation, Numer. Math., Volume 134 (2016) no. 2, pp. 249-274 | DOI | MR | Zbl

[7] Rajib Dutta; Ujjwal Koley; Nils Henrik Risebro Convergence of a higher order scheme for the Korteweg–de Vries equation, SIAM J. Numer. Anal., Volume 53 (2015) no. 4, pp. 1963-1983 | DOI | MR | Zbl

[8] Rajib Dutta; Nils Henrik Risebro A note on the convergence of a Crank–Nicolson scheme for the KdV equation, Int. J. Numer. Anal. Model., Volume 13 (2016) no. 5, pp. 657-675 | MR | Zbl

[9] Rajib Dutta; Tanmay Sarkar Operator splitting for the fractional Korteweg–de Vries equation, Numer. Methods Partial Differ. Equations, Volume 37 (2021) no. 6, pp. 3000-3022 | DOI | MR | Zbl

[10] Germán Fonseca; Felipe Linares; Gustavo Ponce The IVP for the dispersion generalized Benjamin–Ono equation in weighted Sobolev spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) no. 5, pp. 763-790 | DOI | Numdam | MR | Zbl

[11] Sondre Tesdal Galtung Convergence rates of a fully discrete Galerkin scheme for the Benjamin–Ono equation, XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications (Christian Klingenberg; Michael Westdickenberg, eds.) (Springer Proceedings in Mathematics & Statistics), Volume 236, Springer (2016), pp. 589-601 | MR | Zbl

[12] Sondre Tesdal Galtung Convergent Crank–Nicolson Galerkin Scheme for the Benjamin–Ono Equation, Masters thesis, NTNU Norwegian University of Science and Technology (2016)

[13] Sondre Tesdal Galtung A convergent Crank–Nicolson Galerkin scheme for the Benjamin–Ono equation, Discrete Contin. Dyn. Syst., Volume 38 (2018) no. 3, pp. 1243-1268 | DOI | MR | Zbl

[14] Jean Ginibre; Giorgio Velo Commutator expansions and smoothing properties of generalized Benjamin–Ono equations, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 51 (1989) no. 2, pp. 221-229 | Numdam | MR

[15] Jean Ginibre; Giorgio Velo Smoothing properties and existence of solutions for the generalized Benjamin–Ono equation, J. Differ. Equations, Volume 93 (1991) no. 1, pp. 150-212 | DOI | MR

[16] Gerd Grubb Fourier methods for fractional-order operators (2022) | arXiv

[17] Sebastian Herr; Alexandru D. Ionescu; Carlos E. Kenig; Herbert Koch A para-differential renormalization technique for nonlinear dispersive equations, Commun. Partial Differ. Equations, Volume 35 (2010) no. 10, pp. 1827-1875 | DOI | MR | Zbl

[18] Helge Holden; Kenneth Hvistendahl Karlsen; Nils Henrik Risebro Operator Splitting Methods for Generalized Korteweg–de Vries Equations, J. Comput. Phys., Volume 153 (1999) no. 1, pp. 203-222 | DOI | MR | Zbl

[19] Helge Holden; Ujjwal Koley; Nils Henrik Risebro Convergence of a fully discrete finite difference scheme for the Korteweg–de Vries equation, IMA J. Numer. Anal., Volume 35 (2015) no. 3, pp. 1047-1077 | DOI | MR | Zbl

[20] Alexandru D. Ionescu; Carlos E. Kenig Global well-posedness of the Benjamin–Ono equation in low-regularity spaces, J. Am. Math. Soc., Volume 20 (2007) no. 3, pp. 753-798 | DOI | MR | Zbl

[21] Tosio Kato On the Cauchy problem for the (generalized) Korteweg–de Vries equation, Adv. Math., Suppl. Stud., Volume 8 (1983), pp. 93-128 | MR | Zbl

[22] Carlos E. Kenig; Gustavo Ponce; Luis Vega Well-Posedness of the Initial Value Problem for the Korteweg–de Vries Equation, J. Am. Math. Soc., Volume 4 (1991) no. 2, pp. 323-347 | DOI | MR | Zbl

[23] Rowan Killip; Monica Vişan KdV is wellposed in $H^{- 1}$ , Ann. Math., Volume 190 (2019) no. 1, pp. 249-305 | Zbl

[24] Christian Klein; Jean-Claude Saut A numerical approach to blow-up issues for dispersive perturbations of Burgers’ equation, Phys. D: Nonlinear Phenom., Volume 295 (2015), pp. 46-65 | DOI | MR | Zbl

[25] Ujjwal Koley; Deep Ray; Tanmay Sarkar Multilevel Monte Carlo finite difference methods for fractional conservation laws with random data, SIAM/ASA J. Uncertain. Quantif., Volume 9 (2021) no. 1, pp. 65-105 | DOI | MR | Zbl

[26] D. J. Korteweg; Gustav De Vries On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. (5), Volume 39 (1895) no. 240, pp. 422-443 | DOI | MR

[27] Mateusz Kwaśnicki Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., Volume 20 (2017) no. 1, pp. 7-51 | DOI | MR | Zbl

[28] Felipe Linares; Didier Pilod; Jean-Claude Saut Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, SIAM J. Math. Anal., Volume 46 (2014) no. 2, pp. 1505-1537 | DOI | MR | Zbl

[29] Felipe Linares; Gustavo Ponce Introduction to Nonlinear Dispersive Equations, Universitext, Springer, 2014

[30] Luc Molinet; Didier Pilod The Cauchy problem for the Benjamin–Ono equation in $L^{2}$ revisited, Anal. PDE, Volume 5 (2012) no. 2, pp. 365-395 | DOI | MR | Zbl

[31] Luc Molinet; Didier Pilod; Stéphane Vento On well-posedness for some dispersive perturbations of Burgers’ equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 35 (2018) no. 7, pp. 1719-1756 | Numdam | MR | Zbl

[32] Igor Podlubny Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198, Academic Press Inc., 1999

[33] Alfio Quarteroni; Alberto Valli Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, 23, Springer, 2008

[34] Anders Sjöberg On the Korteweg–de Vries equation: existence and uniqueness, J. Math. Anal. Appl., Volume 29 (1970) no. 3, pp. 569-579 | DOI | MR | Zbl

[35] Olaf Steinbach On the stability of the L $^{2}$ projection in fractional Sobolev spaces, Numer. Math., Volume 88 (2001) no. 2, pp. 367-379 | DOI | MR | Zbl

[36] Terence Tao Global well-posedness of the Benjamin–Ono equation in $H^{1} (ℝ)$ , J. Hyperbolic Differ. Equ., Volume 1 (2004) no. 1, pp. 27-49 | MR | Zbl

[37] Vidar Thomée; A. S. Vasudeva Murthy A numerical method for the Benjamin–Ono equation, BIT, Volume 38 (1998), pp. 597-611 | DOI | MR | Zbl

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