A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation

Sondre Tesdal Galtung

doi:10.3934/dcds.2018051

Article Contents

2018, Volume 38, Issue 3: 1243-1268. Doi: 10.3934/dcds.2018051

This issue Previous Article Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system Next Article Weak regularization by stochastic drift : Result and counter example

A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation

Sondre Tesdal Galtung

Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

Received: February 28, 2017

Revised: April 30, 2017

Published: June 2018

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Abstract

In this paper we prove the convergence of a Crank-Nicolson type Galerkin finite element scheme for the initial value problem associated to the Benjamin-Ono equation. The proof is based on a recent result for a similar discrete scheme for the Korteweg-de Vries equation and utilizes a local smoothing effect to bound the $ H^{1/2} $-norm of the approximations locally. This enables us to show that the scheme converges strongly in $ L^{2}(0,T;L^{2}_{\text{loc}}(\mathbb{R})) $ to a weak solution of the equation for initial data in $L^{2}(\mathbb{R})$ and some $ T > 0 $. Finally we illustrate the method with some numerical examples.

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Mathematics Subject Classification: Primary: 35Q53, 65M60; Secondary: 35Q51, 65M12.

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Figure 1. Numerical approximation for $N = 256$ and exact solution for $t = 0$, $90$ and $180$, respectively positioned from left to right in the plot, for initial data $u_{s2}$

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Table 1. Relative $L^{2}$-error, $I_1$, $I_2$ and $I_3$ at $t = 90$ and $t = 180$ for initial data $u_{s2}$

t	N	E	rate_E	I₁	I₂	I₃
90	128	0.01844	-1.45 1.58 0.68 1.16 0.08	3.79×l0^-5	-7.05×l0^-4	5.86×l0^-3
	256	0.05021		-6.61×l0^-6	-3.85×l0^-3	1.65×l0^-2
	512	0.01678		-6.64×l0^-6	-9.96×l0^-4	4.70×l0^-3
	1024	0.01044		-4.64×l0^-6	3.62×l0^-4	-1.57×l0^-3
	2048	0.00467		-3.25×l0^-6	4.16×l0^-5	-5.71×l0^-6
	4096	0.00442		-2.29×l0^-6	4.12×l0^-6	2.07×l0^-4
180	128	0.11959	-1.32 1.75 0.74 2.35 0.89	1.57×l0^-4	-6.45×10^-4	-1.56×l0^-2
	256	0.29755		2.48×l0^-6	-7.80×l0^-3	3.32×l0^-2
	512	0.08869		-3.76×l0^-6	-2.42×l0^-3	1.12×l0^-2
	1024	0.05295		-2.69×10^-6	9.22×l0^-4	-4.11×10^-3
	2048	0.01040		-1.82×10^-6	1.16×10^-4	-4.74×10^-4
	4096	0.00561		-1.26×l0^-6	1.46×10^-5	-1.67×l0^-5

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References

	L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D, 40 (1989), 360–392, URL http://dx.doi.org/10.1016/0167-2789(89)90050-X. doi: 10.1016/0167-2789(89)90050-X.
	J. P. Albert, J. L. Bona and D. B. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves, Phys. D, 24 (1987), 343–366, URL http://dx.doi.org/10.1016/0167-2789(87)90084-4. doi: 10.1016/0167-2789(87)90084-4.
	T. B. Benjamin , Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967) , 559-592. doi: 10.1017/S002211206700103X.
	T. L. Bock and M. D. Kruskal, A two-parameter Miura transformation of the Benjamin– Ono equation, Phys. Lett. A, 74 (1979), 173–176, URL http://dx.doi.org/10.1016/0375-9601(79)90762-X. doi: 10.1016/0375-9601(79)90762-X.
	P. G. Ciarlet, The Finite Element Method for Elliptic Problems, vol. 40 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002, http://dx.doi.org/10.1137/1.9780898719208.
	Z. Deng and H. Ma, Optimal error estimates of the Fourier spectral method for a class of nonlocal, nonlinear dispersive wave equations, Appl. Numer. Math., 59 (2009), 988–1010, URL http://dx.doi.org/10.1016/j.apnum.2008.03.042. doi: 10.1016/j.apnum.2008.03.042.
	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573, URL http://dx.doi.org/10.1016/j.bulsci.2011.12.004. doi: 10.1016/j.bulsci.2011.12.004.
	V. A. Dougalis, A. Duran and D. Mitsotakis, Numerical solution of the Benjamin equation, Wave Motion, 52 (2015), 194–215, URL http://dx.doi.org/10.1016/j.wavemoti.2014.10.004.
	R. Dutta, H. Holden, U. Koley and N. H. Risebro, Operator splitting for the Benjamin–Ono equation, J. Differential Equations, 259 (2015), 6694–6717, URL http://dx.doi.org/10.1016/j.jde.2015.08.002. doi: 10.1016/j.jde.2015.08.002.
	R. Dutta, H. Holden, U. Koley and N. H. Risebro, Convergence of finite difference schemes for the Benjamin–Ono equation, Numer. Math., 134 (2016), 249–274, URL http://dx.doi.org/10.1007/s00211-015-0778-6. doi: 10.1007/s00211-015-0778-6.
	R. Dutta, U. Koley and N. H. Risebro, Convergence of a higher order scheme for the Korteweg–de Vries equation, SIAM J. Numer. Anal., 53 (2015), 1963–1983, URL http://dx.doi.org/10.1137/140982532.
	R. Dutta and N. H. Risebro , A note on the convergence of a Crank-Nicolson scheme for the KdV equation, Int. J. Numer. Anal. Model., 13 (2016) , 657-675.
	A. S. Fokas and M. J. Ablowitz, The inverse scattering transform for the Benjamin–Ono equation—a pivot to multidimensional problems, Stud. Appl. Math., 68 (1983), 1–10, URL http://dx.doi.org/10.1002/sapm19836811. doi: 10.1002/sapm19836811.
	S. T. Galtung, Convergence rates of a fully discrete Galerkin scheme for the Benjamin-Ono equation, to appear in Springer Proceedings in Mathematics and Statistics, arXiv:1611.09041, URL http://adsabs.harvard.edu/abs/2016arXiv161109041T.
	S. T. Galtung, A Convergent Crank-Nicolson Galerkin Scheme for the Benjamin-Ono Equation, Master's thesis, NTNU Norwegian University of Science and Technology, 2016, URL http://hdl.handle.net/11250/2395092.
	J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin–Ono equation, J. Differential Equations, 93 (1991), 150–212, URL http://dx.doi.org/10.1016/0022-0396(91)90025-5. doi: 10.1016/0022-0396(91)90025-5.
	J. Ginibre and G. Velo, Commutator expansions and smoothing properties of generalized Benjamin–Ono equations, Ann. Inst. H. Poincaré Phys. Théor., 51 (1989), 221–229, URL http://www.numdam.org/item?id=AIHPA_1989__51_2_221_0.
	L. Grafakos, Classical Fourier Analysis, vol. 249 of Graduate Texts in Mathematics, 3rd edition, Springer, New York, 2014, http://dx.doi.org/10.1007/978-1-4939-1194-3.
	H. Holden, U. Koley and N. H. Risebro, Convergence of a fully discrete finite difference scheme for the Korteweg–de Vries equation, IMA J. Numer. Anal., 35 (2015), 1047–1077, URL http://dx.doi.org/10.1093/imanum/dru040. doi: 10.1093/imanum/dru040.
	A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin–Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753–798 (electronic), URL http://dx.doi.org/10.1090/S0894-0347-06-00551-0. doi: 10.1090/S0894-0347-06-00551-0.
	R. J. Iório Jr., On the Cauchy problem for the Benjamin–Ono equation, Comm. Partial Differential Equations, 11 (1986), 1031–1081, URL http://dx.doi.org/10.1080/03605308608820456. doi: 10.1080/03605308608820456.
	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, in Studies in applied mathematics, vol. 8 of Adv. Math. Suppl. Stud., Academic Press, New York, 1983, 93-128.
	D. J. Kaup and Y. Matsuno, The inverse scattering transform for the Benjamin–Ono equation, Stud. Appl. Math., 101 (1998), 73–98, URL http://dx.doi.org/10.1111/1467-9590.00086. doi: 10.1111/1467-9590.00086.
	F. W. King, Hilbert Transforms. Vol. 1, vol. 124 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2009.
	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edition, Universitext, Springer, New York, 2015, URL http://dx.doi.org/10.1007/978-1-4939-2181-2.
	H. Ono , Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975) , 1082-1091. doi: 10.1143/JPSJ.39.1082.
	B. Pelloni and V. A. Dougalis, Numerical solution of some nonlocal, nonlinear dispersive wave equations, J. Nonlinear Sci., 10 (2000), 1–22, URL http://dx.doi.org/10.1007/s003329910001. doi: 10.1007/s003329910001.
	T. Tao, Global well-posedness of the Benjamin–Ono equation in H¹(R), J. Hyperbolic Differ. Equ., 1 (2004), 27–49, URL http://dx.doi.org/10.1142/S0219891604000032. doi: 10.1142/S0219891604000032.
	V. Thomée and A. S. Vasudeva Murthy, A numerical method for the Benjamin–Ono equation, BIT, 38 (1998), 597–611, URL http://dx.doi.org/10.1007/BF02510262. doi: 10.1007/BF02510262.