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

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

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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

R. Dutta, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

F. W. King, Hilbert Transforms. Vol. 1, vol. 124 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2009.

[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.

[26]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

R. Dutta, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

F. W. King, Hilbert Transforms. Vol. 1, vol. 124 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2009.

[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.

[26]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. doi: 10.1143/JPSJ.39.1082.

[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.

[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.

[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.

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 $u_{s2}$
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 rateE 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
t N E rateE 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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