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

Sondre Tesdal Galtung

doi:10.3934/dcds.2018051

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March 2018, 38(3): 1243-1268. doi: 10.3934/dcds.2018051

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 March 2017 Revised May 2017 Published December 2017

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.

Keywords: Benjamin-Ono equation, compactness, convergence, Crank-Nicolson, finite element method.

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

Citation: Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051

References:

[1]	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. Google Scholar
[2]	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. Google Scholar
[3]	T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X. Google Scholar
[4]	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. Google Scholar
[5]	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. Google Scholar
[6]	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. Google Scholar
[7]	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. Google Scholar
[8]	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. Google Scholar
[9]	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. Google Scholar
[10]	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. Google Scholar
[11]	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. Google Scholar
[12]	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. Google Scholar
[13]	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. Google Scholar
[14]	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. Google Scholar
[15]	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. Google Scholar
[16]	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. Google Scholar
[17]	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. Google Scholar
[18]	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. Google Scholar
[19]	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. Google Scholar
[20]	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. Google Scholar
[21]	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. Google Scholar
[22]	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. Google Scholar
[23]	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. Google Scholar
[24]	F. W. King, Hilbert Transforms. Vol. 1, vol. 124 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2009. Google Scholar
[25]	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. Google Scholar
[26]	H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082. Google Scholar
[27]	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. Google Scholar
[28]	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. Google Scholar
[29]	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. Google Scholar

show all references

References:

[1]	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. Google Scholar
[2]	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. Google Scholar
[3]	T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X. Google Scholar
[4]	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. Google Scholar
[5]	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. Google Scholar
[6]	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. Google Scholar
[7]	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. Google Scholar
[8]	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. Google Scholar
[9]	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. Google Scholar
[10]	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. Google Scholar
[11]	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. Google Scholar
[12]	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. Google Scholar
[13]	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. Google Scholar
[14]	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. Google Scholar
[15]	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. Google Scholar
[16]	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. Google Scholar
[17]	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. Google Scholar
[18]	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. Google Scholar
[19]	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. Google Scholar
[20]	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. Google Scholar
[21]	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. Google Scholar
[22]	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. Google Scholar
[23]	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. Google Scholar
[24]	F. W. King, Hilbert Transforms. Vol. 1, vol. 124 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2009. Google Scholar
[25]	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. Google Scholar
[26]	H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082. Google Scholar
[27]	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. Google Scholar
[28]	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. Google Scholar
[29]	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. Google Scholar

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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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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