Stability and convergence analysis of a Crank-Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation. (English) Zbl 07883472
Summary: 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 \(H^{\alpha/2}\)-norm of the approximated solution, where \(\alpha\in[1, 2)\). We demonstrate that the scheme converges strongly in \(L^2(0, T; L^2_{loc}(\mathbb{R}))\) to a weak solution of the fractional KdV equation provided the initial data in \(L^2(\mathbb{R})\). 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.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35D30 | Weak solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 41A25 | Rate of convergence, degree of approximation |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
fractional Korteweg-de Vries equation; fractional Laplacian; Galerkin method; rate of convergence; \(L^2\) initial dataReferences:
| [1] | Abdelouhab, Laziz; Bona, Jerry L.; Felland, M.; Saut, Jean-Claude, Nonlocal models for nonlinear, dispersive waves, Physica D, 40, 3, 360-392, 1989 · Zbl 0699.35227 · doi:10.1016/0167-2789(89)90050-X |
| [2] | Biccari, Umberto; Warma, Mahamadi; Zuazua, Enrique, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17, 2, 387-409, 2017 · Zbl 1360.35033 · doi:10.1515/ans-2017-0014 |
| [3] | Ciarlet, Philippe G., The Finite Element Method for Elliptic Problems, 2002, Society for Industrial and Applied Mathematics · Zbl 0999.65129 · doi:10.1137/1.9780898719208 |
| [4] | Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 5, 521-573, 2012 · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004 |
| [5] | Dutta, Rajib; Holden, Helge; Koley, Ujjwal; Risebro, Nils Henrik, Operator splitting for the Benjamin-Ono equation, J. Differ. Equations, 259, 11, 6694-6717, 2015 · Zbl 1326.35311 · doi:10.1016/j.jde.2015.08.002 |
| [6] | Dutta, Rajib; Holden, Helge; Koley, Ujjwal; Risebro, Nils Henrik, Convergence of finite difference schemes for the Benjamin-Ono equation, Numer. Math., 134, 2, 249-274, 2016 · Zbl 1353.65090 · doi:10.1007/s00211-015-0778-6 |
| [7] | Dutta, Rajib; Koley, Ujjwal; Risebro, Nils Henrik, Convergence of a higher order scheme for the Korteweg-de Vries equation, SIAM J. Numer. Anal., 53, 4, 1963-1983, 2015 · Zbl 1342.35301 · doi:10.1137/140982532 |
| [8] | Dutta, Rajib; Risebro, Nils Henrik, A note on the convergence of a Crank-Nicolson scheme for the KdV equation, Int. J. Numer. Anal. Model., 13, 5, 657-675, 2016 · Zbl 1401.65104 |
| [9] | Dutta, Rajib; Sarkar, Tanmay, Operator splitting for the fractional Korteweg-de Vries equation, Numer. Methods Partial Differ. Equations, 37, 6, 3000-3022, 2021 · Zbl 07775888 · doi:10.1002/num.22810 |
| [10] | Fonseca, Germán; Linares, Felipe; Ponce, Gustavo, The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 30, 5, 763-790, 2013 · Zbl 1511.35035 · doi:10.1016/j.anihpc.2012.06.006 |
| [11] | Galtung, Sondre Tesdal; Klingenberg, Christian; Westdickenberg, Michael, Convergence rates of a fully discrete Galerkin scheme for the Benjamin-Ono equation, XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications, 236, 589-601, 2016 · Zbl 1405.65119 |
| [12] | Galtung, Sondre Tesdal, Convergent Crank-Nicolson Galerkin Scheme for the Benjamin-Ono Equation, 2016 · Zbl 1397.35252 |
| [13] | Galtung, Sondre Tesdal, A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation, Discrete Contin. Dyn. Syst., 38, 3, 1243-1268, 2018 · Zbl 1397.35252 · doi:10.3934/dcds.2018051 |
| [14] | Ginibre, Jean; Velo, Giorgio, Commutator expansions and smoothing properties of generalized Benjamin-Ono equations, Ann. Inst. Henri Poincaré, Phys. Théor., 51, 2, 221-229, 1989 · Zbl 0705.35126 |
| [15] | Ginibre, Jean; Velo, Giorgio, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation, J. Differ. Equations, 93, 1, 150-212, 1991 · Zbl 0770.35063 · doi:10.1016/0022-0396(91)90025-5 |
| [16] | Grubb, Gerd, Fourier methods for fractional-order operators, 2022 · Zbl 1476.35309 |
| [17] | Herr, Sebastian; Ionescu, Alexandru D.; Kenig, Carlos E.; Koch, Herbert, A para-differential renormalization technique for nonlinear dispersive equations, Commun. Partial Differ. Equations, 35, 10, 1827-1875, 2010 · Zbl 1214.35089 · doi:10.1080/03605302.2010.487232 |
| [18] | Holden, Helge; Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik, Operator Splitting Methods for Generalized Korteweg-de Vries Equations, J. Comput. Phys., 153, 1, 203-222, 1999 · Zbl 0947.65102 · doi:10.1006/jcph.1999.6273 |
| [19] | Holden, Helge; Koley, Ujjwal; Risebro, Nils Henrik, Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation, IMA J. Numer. Anal., 35, 3, 1047-1077, 2015 · Zbl 1348.65121 · doi:10.1093/imanum/dru040 |
| [20] | Ionescu, Alexandru D.; Kenig, Carlos E., Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Am. Math. Soc., 20, 3, 753-798, 2007 · Zbl 1123.35055 · doi:10.1090/S0894-0347-06-00551-0 |
| [21] | Kato, Tosio, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Adv. Math., Suppl. Stud., 8, 93-128, 1983 · Zbl 0549.34001 |
| [22] | Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis, Well-Posedness of the Initial Value Problem for the Korteweg-de Vries Equation, J. Am. Math. Soc., 4, 2, 323-347, 1991 · Zbl 0737.35102 · doi:10.1090/S0894-0347-1991-1086966-0 |
| [23] | Killip, Rowan; Vişan, Monica, KdV is wellposed in \({H}^{-1}\), Ann. Math., 190, 1, 249-305, 2019 · Zbl 1426.35203 |
| [24] | Klein, Christian; Saut, Jean-Claude, A numerical approach to blow-up issues for dispersive perturbations of Burgers’ equation, Phys. D: Nonlinear Phenom., 295, 46-65, 2015 · Zbl 1364.35047 · doi:10.1016/j.physd.2014.12.004 |
| [25] | Koley, Ujjwal; Ray, Deep; Sarkar, Tanmay, Multilevel Monte Carlo finite difference methods for fractional conservation laws with random data, SIAM/ASA J. Uncertain. Quantif., 9, 1, 65-105, 2021 · Zbl 1471.65104 · doi:10.1137/19M1279447 |
| [26] | Korteweg, D. J.; De Vries, Gustav, 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), 39, 240, 422-443, 1895 · JFM 26.0881.02 · doi:10.1080/14786449508620739 |
| [27] | Kwaśnicki, Mateusz, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20, 1, 7-51, 2017 · Zbl 1375.47038 · doi:10.1515/fca-2017-0002 |
| [28] | Linares, Felipe; Pilod, Didier; Saut, Jean-Claude, Dispersive perturbations of Burgers and hyperbolic equations I: Local theory, SIAM J. Math. Anal., 46, 2, 1505-1537, 2014 · Zbl 1294.35124 · doi:10.1137/130912001 |
| [29] | Linares, Felipe; Ponce, Gustavo, Introduction to Nonlinear Dispersive Equations, 2014, Springer |
| [30] | Molinet, Luc; Pilod, Didier, The Cauchy problem for the Benjamin-Ono equation in \({L}^2\) revisited, Anal. PDE, 5, 2, 365-395, 2012 · Zbl 1273.35096 · doi:10.2140/apde.2012.5.365 |
| [31] | Molinet, Luc; Pilod, Didier; Vento, Stéphane, On well-posedness for some dispersive perturbations of Burgers’ equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 35, 7, 1719-1756, 2018 · Zbl 1459.76024 |
| [32] | Podlubny, Igor, Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, 198, 1999, Academic Press Inc. · Zbl 0924.34008 |
| [33] | Quarteroni, Alfio; Valli, Alberto, Numerical approximation of partial differential equations, 23, 2008, Springer · Zbl 1151.65339 |
| [34] | Sjöberg, Anders, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl., 29, 3, 569-579, 1970 · Zbl 0179.43101 · doi:10.1016/0022-247X(70)90068-5 |
| [35] | Steinbach, Olaf, On the stability of the \(L^2\) projection in fractional Sobolev spaces, Numer. Math., 88, 2, 367-379, 2001 · Zbl 0989.65124 · doi:10.1007/PL00005449 |
| [36] | Tao, Terence, Global well-posedness of the Benjamin-Ono equation in \({H}^1(\mathbb{R})\), J. Hyperbolic Differ. Equ., 1, 1, 27-49, 2004 · Zbl 1055.35104 |
| [37] | Thomée, Vidar; Vasudeva Murthy, A. S., A numerical method for the Benjamin-Ono equation, BIT, 38, 597-611, 1998 · Zbl 0916.65139 · doi:10.1007/BF02510262 |
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