×

Energy-conserving Hamiltonian boundary value methods for the numerical solution of the Korteweg-de Vries equation. (English) Zbl 1414.65024

The authors investigate the solutions of the well-known KdV equation in the case of periodic boundary conditions. To show the efficiency of the solutions, the authors apply a Fourier-Galerkin semidiscretization in space. This yields in a first step a Hamiltonian ODE problem. Next, in a second step, by observing an energy conservation in the Hamiltonian boundary value problem, the authors develop solutions in time by means of an energy-conserving method evolving Sylvester-type equations. The efficiency of the method is improved by the implementation of several numerical examples.
Overall, it is an interesting work as it considers a famous problem of understanding the KdV equation which yields automatically an advance in understanding related problems such as wave propagation on the surface of shallow water, solitons, etc.
The list of references is somehow exhaustive and reflects a good literature on the problem and the good background of the authors.
I think finally that this work may open important directions by considering it on more complicated spaces such as fractal domains where a suitable notion of differentiablility, energy, periodicity should be defined.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
35C08 Soliton solutions
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kappeler, T.; Pöschel, J., On the well-posedness of the periodic KdV equation in high regularity classes, (Craig, W., Hamiltonian Systems and Applications (2008), Springer), 431-441 · Zbl 1251.35130
[2] Zabusky, N. L.; Kruskal, M. D., Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15, 240-243 (1965) · Zbl 1201.35174
[3] Bambusi, D.; Carati, A.; Maiocchi, A.; Maspero, A., Some analytic results on the FPU paradox, (Guyenne, P.; Nicholls, D.; Sulem, C., Hamiltonian Partial Differential Equations and Applications, vol. 75 (2015), Fields Institute Communications), 235-254 · Zbl 1339.37074
[4] Gardner, C. S., Korteweg-de Vries equation and generalization. IV. The Korteweg-de Vries equation as a Hamiltonian system, J. Math. Phys., 12, 8, 1548-1651 (1971) · Zbl 0283.35021
[5] Bättig, D.; Kappeler, T.; Mityagin, B., On the Korteweg-de Vries equation: Frequencies and initial value problem, Pacific J. Math., 181, 1, 1-5 (1997) · Zbl 0899.35096
[6] Olver, P. J., Hamiltonian and non-Hamiltonian models for water wawes, Trends and Applications of Pure Mathematics to Mechanics, 273-290 (1984)
[7] Nahas, J.; Ponce, G., On the persistence properties of solutions of nonlinear dispersive equations, (Weighted Sobolev Spaces. Harmonic Analysis and Nonlinear Partial Differential Equations (2011), Res. Inst. Math. Sci. (RIMS): Res. Inst. Math. Sci. (RIMS) Kyoto), 23-36 · Zbl 1236.35169
[8] Isaza, P.; Linares, F.; Ponce, G., On decay properties of solutions of the \(k\)-generalized KdV equation, Comm. Math. Phys., 324, 129-146 (2013) · Zbl 1284.35373
[9] Guan, H.; Kuksin, S., The KdV equation under periodic boundary conditions and its perturbations, Nonlinearity, 27, 9, R61-R88 (2014) · Zbl 1301.35132
[10] Isaza, P.; Linares, F.; Ponce, G., On the propagation of regularity and decay of solutions to the \(k\)-generalized Korteweg-de Vries equation, Comm. Partial Differential Equations, 40, 1336-1364 (2015) · Zbl 1326.35317
[11] Tappert, F., Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method, (Newell, A. C., Nonlinear Wave Motion (1974), American Mathematical Society: American Mathematical Society Providence), 215-216
[12] Alexander, M. E.; Morris, J. L., Galerkin methods applied to some model equations for non-linear dispersive waves, J. Comput. Phys., 30, 3, 428-451 (1979) · Zbl 0407.76014
[13] Winther, R., A conservative finite element method for the Korteweg-de Vries equation, Math. Comp., 34, 149, 23-43 (1980) · Zbl 0422.65063
[14] Bona, J. L.; Dougalis, V. A.; Karakashian A., O., Fully discrete Galerkin methods for the Korteweg-de Vries equation, Comput. Math. Appl. A, 12, 7, 859-884 (1986) · Zbl 0597.65072
[15] Huang, M., A Hamiltonian approximation to simulate solitary waves of the Korteweg-de Vries equation, Math. Comp., 56, 194, 607-620 (1991) · Zbl 0723.65100
[16] de Frutos, J.; Sanz-Sern, J. M., Accuracy and conservation properties in numerical integration: The case of the Korteweg-de Vries equation, Numer. Math., 75, 421-445 (1997) · Zbl 0876.65068
[17] Yan, J.; Shu, C.-W., A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40, 2, 769-791 (2002) · Zbl 1021.65050
[18] Liu, H.; Yan, J., A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect, J. Comput. Phys., 215, 197-218 (2006) · Zbl 1092.65083
[19] Xu, Y.; Shu, C. W., Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations, Comput. Methods Appl. Mech. Engrg., 196, 37-40, 3805-3822 (2007) · Zbl 1173.65338
[20] Bona, J. L.; Chen, H.; Karakashian, O.; Xing, Y., Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation, Math. Comp., 82, 283, 1401-1432 (2013) · Zbl 1276.65058
[21] Hufford, C.; Xing, Y., Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation, J. Comput. Appl. Math., 255, 441-455 (2014) · Zbl 1291.65301
[22] Chen, Y.; Cockburn, B.; Dong, B., Superconvergent HDG methods for linear, stationary, third-order equations in one-space dimension, Math. Comp., 85, 302, 2715-2742 (2016) · Zbl 1344.65070
[23] Ascher, U.; McLachlan, R., Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48, 255-269 (2004) · Zbl 1038.65138
[24] Zhao, P.; Qin, M., Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation, J. Phys. A: Math. Gen., 33, 3613-3626 (2006) · Zbl 0989.37062
[25] Li, X.; Zhang, L.; Wang, S., A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator, Appl. Math. Comput., 219, 3187-3197 (2012) · Zbl 1309.65099
[26] Holden, H.; Karlsen, K. H.; Risebro, N. H.; Tao, T., Operator splitting methods for the Korteweg-de Vries equation, Math. Comp., 80, 821-846 (2011) · Zbl 1219.35235
[27] Hofmanová, M.; Schratz, K., An exponential-type integrator for the KdV equation, Numer. Math., 136, 4, 1117-1137 (2017) · Zbl 1454.65034
[28] Wang, J.; Wang, Y., Local structure-preserving algorithms for the KdV equation, J. Comput. Math., 35, 3, 289-318 (2017) · Zbl 1413.65459
[29] Cui, Y.; Mao, D.-k., Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation, J. Comput. Phys., 227, 376-399 (2007) · Zbl 1131.65073
[30] Karasözen, B.; Şimşek, G., Energy preserving integration of bi-Hamiltonian partial differential equations, Appl. Math. Lett., 26, 1125-1133 (2013) · Zbl 1308.35249
[31] Liu, H.; Yi, N., A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation, J. Comput. Phys., 321, 776-796 (2016) · Zbl 1349.65462
[32] Jackaman, J.; Papamikos, G.; Pryer, T., The design of conservative finite element discretisations for the vectorial modified KdV equation (2017), arXiv:1710.03527 [math.NA]
[33] Song, M. Z.; Qian, X.; Zhang, H.; Song, S. H., Hamiltonian Boundary Value Method for the nonlinear Schrödinger equation and the Korteweg-de Vries equation, Adv. Appl. Math. Mech., 9, 868-886 (2017) · Zbl 1488.65518
[34] Giusti, E., Direct Methods in the Calculus of Variations (2003), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge, NJ · Zbl 1028.49001
[35] Leimkuhler, B.; Reich, S., Simulating Hamiltonian Dynamics (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1069.65139
[36] Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1094.65125
[37] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0658.76001
[38] Brugnano, L.; Frasca Caccia, G.; Iavernaro, F., Energy conservation issues in the numerical solution of the semilinear wave equation, Appl. Math. Comput., 270, 842-870 (2015) · Zbl 1410.65477
[39] Dahlquist, G.; Bijörk, Å., Numerical Methods in Scientific Computing, vol. 1 (2008), SIAM: SIAM Philadelphia · Zbl 1153.65001
[40] Sanz-Serna, J. M.; Calvo, M. P., Numerical Hamiltonian Problems (1994), Chapman & Hall: Chapman & Hall London · Zbl 0816.65042
[41] Brugnano, L.; Iavernaro, F., Line Integral Methods for Conservative Problems (2016), Chapman et Hall/CRC: Chapman et Hall/CRC Boca Raton, FL · Zbl 1335.65097
[42] Brugnano, L.; Iavernaro, F.; Trigiante, D., Hamiltonian BVMs (HBVMs): A family of “drift-free” methods for integrating polynomial Hamiltonian systems, AIP Conf. Proc., 1168, 715-718 (2009) · Zbl 1182.65188
[43] Brugnano, L.; Iavernaro, F.; Trigiante, D., Hamiltonian Boundary Value Methods (energy preserving discrete line integral methods), JNAIAM. J. Numer. Anal. Ind. Appl. Math., 5, 1-2, 17-37 (2010) · Zbl 1432.65182
[44] Brugnano, L.; Iavernaro, F.; Trigiante, D., A note on the efficient implementation of Hamiltonian BVMs, J. Comput. Appl. Math., 236, 375-383 (2011) · Zbl 1228.65107
[45] Brugnano, L.; Iavernaro, F.; Trigiante, D., A simple framework for the derivation and analysis of effective one-step methods for ODEs, Appl. Math. Comput., 218, 8475-8485 (2012) · Zbl 1245.65086
[46] Brugnano, L.; Frasca Caccia, G.; Iavernaro, F., Efficient implementation of Gauss collocation and Hamiltonian Boundary Value Methods, Numer. Algorithms, 65, 633-650 (2014) · Zbl 1291.65357
[47] Brugnano, L.; Iavernaro, F.; Trigiante, D., Analisys of Hamiltonian Boundary Value Methods, HBVMs: A class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems, Commun. Nonlinear Sci. Numer. Simul., 20, 650-667 (2015) · Zbl 1304.65262
[48] Barletti, L.; Brugnano, L.; Frasca Caccia, G.; Iavernaro, F., Energy-conserving methods for the nonlinear Schrödinger equation, Appl. Math. Comput., 318, 3-18 (2018) · Zbl 1426.65202
[49] Quispel, G. R.W.; McLaren, D. I., A new class of energy-preserving numerical integration methods, J. Phys. A, 41, Article 045206 pp. (2008), (7pp) · Zbl 1132.65065
[50] Brugnano, L.; Magherini, C., Blended implementation of block implicit methods for ODEs, Appl. Numer. Math., 42, 29-45 (2002) · Zbl 1006.65078
[51] Brugnano, L.; Trigiante, D., Boundary value methods: The third way between linear multistep and Runge-Kutta methods, Comput. Math. Appl., 36, 10-12, 269-284 (1998) · Zbl 0933.65082
[52] Brugnano, L.; Magherini, C., Recent advances in linear analysis of convergence for splittings for solving ODE problems, Appl. Numer. Math., 59, 542-557 (2009) · Zbl 1162.65039
[53] Brugnano, L.; Magherini, C., The BiM code for the numerical solution of ODEs, J. Comput. Appl. Math., 164-165, 145-158 (2004) · Zbl 1038.65063
[54] Brugnano, L.; Magherini, C.; Mugnai, F., Blended implicit methods for the numerical solution of DAE problems, J. Comput. Appl. Math., 189, 34-50 (2006) · Zbl 1088.65076
[55] https://archimede.dm.uniba.it/testset/testsetivpsolvers/; https://archimede.dm.uniba.it/testset/testsetivpsolvers/
[56] Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), The Johns Hopkins Univeristy Press: The Johns Hopkins Univeristy Press Baltimore · Zbl 0865.65009
[57] Driscoll, T. A.; Hale, N.; Trefethen, L. N., Chebfun Guide (2014), Pafnuty Publications: Pafnuty Publications Oxford, UK, URL: http://www.chebfun.org
[58] http://www.chebfun.org/examples/pde/KdV.html; http://www.chebfun.org/examples/pde/KdV.html
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.