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Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation. (English) Zbl 1131.65073
We develop a finite-volume scheme for the Korteweg de Vries equation which conserves both the momentum and energy. The main ingredient of the method is a numerical device we developed in recent years that enables us to construct a numerical method for a partial differential equation that also simulates its related equations. In the method, numerical approximations to both the momentum and energy are conservatively computed. The operator splitting approach is adopted in constructing the method in which the conservation and dispersion parts of the equation are alternatively solved; our numerical device is applied in solving the conservation part of the equation. The feasibility and stability of the method is discussed, which involves an important property of the method, the so-called Jensen condition. The truncation error of the method is analyzed, which shows that the method is second-order accurate. Finally, several numerical examples, including the Zabusky-Kruskal’s example, are presented to show the good stability property of the method for long-time numerical integration.

MSC:
65M06Finite difference methods (IVP of PDE)
35Q53KdV-like (Korteweg-de Vries) equations
65M12Stability and convergence of numerical methods (IVP of PDE)
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References:
[1] Ascher, U.; Mclachlan, R.: Multisymplectic box schemes and the Korteweg -- de Vries equation. Appl. numer. Math. 48, 255-269 (2004) · Zbl 1038.65138
[2] Ascher, U.; Mclachlan, R.: On symplectic and multisymplectic schemes for the KdV equation. J. sci. Comput. 25, 83-104 (2005) · Zbl 1203.65277
[3] Bridges, T.; Reich, S.: Numerical methods for hamiltonia pdes. J. phys. A: math. Gen. 39, 5287-5320 (2006) · Zbl 1090.65138
[4] Y. Cui, Numerical scheme satisfying two conservation laws for KdV equation, Master’s thesis, No. 11903 -- 02720653, Shanghai University, 2005 (in Chinese).
[5] Z. Deng, H. Ma, Optimal error estimates of the Fourier spectral method for the KdV equation, Appl. Math. Comput., Submitted for publication. · Zbl 1187.65106
[6] Z. Deng, Fourier spectral methods for a class of nonlocal, nonlinear dispersive wave equations, Doctoral thesis, No. 11903 -- 04810040 Shanghai University, 2007 (in Chinese).
[7] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. comput. Phys. 71, 231-303 (1987) · Zbl 0652.65067
[8] Holden, H.; Hvistendahl, K.; Risebro, N.: Operator splitting methods for generalized Korteweg -- de Vries equations. J. comput. Phys. 153, 203-222 (1999) · Zbl 0947.65102
[9] Leveque, R. J.: Finite volume methods for hyperbolic problems. (2002) · Zbl 1010.65040
[10] Leveque, R. J.: Numerical methods for conservation laws. (1990) · Zbl 0723.65067
[11] H. Li, Z. Wang, D. Mao, Numerically neither dissipative nor compressive scheme for linear advection equation and its application to the Euler system, J. Sci. Comput., submitted for publication. · Zbl 1203.65140
[12] H. Li, Entropy dissipating scheme for hyperbolic system of conservation laws in one space dimension, Doctoral thesis, No. 11903-02820022, Shanghai University, 2005 (in Chinese).
[13] H. Li, Second-order entropy dissipation scheme for scalar conservation laws in one space dimension, Master’s thesis, No. 11903-99118086, Shanghai University, 2002 (in Chinese).
[14] Li, H.; Mao, D.: The design of the entropy dissipator of the entropy dissipating scheme for scalar conservation law. Chin. J. Comput. phys. 21, 319-326 (2004)
[15] Morton, K.; Mayers, D.: Numerical solution of partial differential equations. (2005) · Zbl 1126.65077
[16] Nouri, F.; Sloan, D.: A comparison of Fourier pseudospectral methods for the solution of the Korteweg -- de Vries equation. J. comput. Phys. 83, 324-344 (1989) · Zbl 0683.65103
[17] Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. anal. 5, 506-517 (1968) · Zbl 0184.38503
[18] Takahashi, R.; Ohkawa, T.: Numerical experiment on interaction of solitons describing recurrence of initial data. Comput. mech. 5, 273-281 (1989)
[19] Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta numerica, 451-512 (2003) · Zbl 1046.65078
[20] Wang, Y.; Wang, B.; Chen, X.: Multisymplectic Euler box scheme for the KdV equation. Chin. phys. Lett. 24, 312-314 (2007)
[21] Z. Wang, Finite difference schemes satisfying multiconservation laws for linear advection equations, Master’s thesis, No. 11903-99118086, Shanghai University, 2006 (in Chinese).
[22] Wang, Z.; Mao, D.: Conservative difference scheme satisfying three conservation laws for linear advection equation. J. shu 12, No. 6, 588-592 (2006) · Zbl 1110.65086
[23] Xu, Y.; Shu, C.: Local discontinuous Galerkin methods for three classes of nonlinear wave equations. J. comput. Math. 22, 250-274 (2004) · Zbl 1050.65093
[24] Yan, J.; Shu, C.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. anal. 40, 769-791 (2002) · Zbl 1021.65050
[25] Zabusky, N. J.; Kruskal, M. D.: Interactions of ”solitons” in a collisionless plasma and the recurrence of initial states. Phys. rev. Lett. 15, 240-243 (1965) · Zbl 1201.35174
[26] 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