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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March
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 |
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.
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. |
| [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 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.
|
| [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 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.
|
| [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. |
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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