Abstract
We examine some symplectic and multisymplectic methods for the notorious Korteweg–de Vries equation, with the question whether the added structure preservation that these methods offer is key in providing high quality schemes for the long time integration of nonlinear, conservative partial differential equations. Concentrating on second order discretizations, several interesting schemes are constructed and studied. Our essential conclusions are that it is possible to design very stable, conservative difference schemes for the nonlinear, conservative KdV equation. Among the best of such schemes are methods which are symplectic or multisymplectic. Semi-explicit, symplectic schemes can be very effective in many situations. Compact box schemes are effective in ensuring that no artificial wiggles appear in the approximate solution. A family of box schemes is constructed, of which the multisymplectic box scheme is a prominent member, which are particularly stable on coarse space–time grids
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U.M. Ascher R.I. McLachlan (2004) ArticleTitleMultisymplectic box schemes and the Korteweg-de Vries equation Appl. Numer. Algorithms. 48 255–269 Occurrence Handle2056917
T.J. Bridges (1997) ArticleTitleMulti-symplectic structures and wave propagation Math. Proc. Camb. Phil. Soc. 121 147–190 Occurrence Handle0892.35123 Occurrence Handle98d:58054 Occurrence Handle10.1017/S0305004196001429
T.J. Bridges S. Reich (2001) ArticleTitleMulti-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity Phys. Lett. A 284 IssueID4–5 184–193 Occurrence Handle2002g:65166
C.J. Budd M.D. Piggott (2003) Geometric Integration and its Applications P.G. Ciarlet F Cucker (Eds) Handbook of Numerical Analysis vol. XI. North-Holland Amsterdam 35–139
J.-P. Croisille (2002) ArticleTitleKeller’s box-scheme for the one-dimensional stationary convection-diffusion equation Computing 68 IssueID1 37–63 Occurrence Handle1006.65120 Occurrence Handle2003c:65103 Occurrence Handle10.1007/s006070200002
P.G. Drazin R.S. Johnson (1989) Solitons: An Introduction Cambridge University Press Cambridge, London
Furihata D. (1999) ArticleTitleFinite difference schemes for ∂u / ∂t (∂/∂x)α δg / δu that inherit energy conservation or dissipation property J. Comp. Phys. 156 181–205
B. Gustafsson H.-O. Kreiss J. Oliger (1995) Time Dependent Problems and Difference Methods Wiley and Sons New York
Hairer E., Lubich C., Wanner G. Geometric Numerical Integration. Springer, Berlin
T.Y. Hou P.D. Lax (1991) ArticleTitleDispersive approximation in fluid dynamics Comm. Pure Appl. Math. 44 1–40 Occurrence Handle91m:76088
Keller, H.B. (1971). A new difference scheme for parabolic problems. In Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) Academic Press, New York. pp. 327–350.
H.-O. Kreiss (1964) ArticleTitleOn difference approximations of the dissipative type for hyperbolic differential equations Comm. Pure Appl. Math. 17 335–353 Occurrence Handle0279.35059 Occurrence Handle29 #4210
Lax P.D., Levermore C.D. (1983). The small dispersion limit of the Korteweg-de Vries equation. Comm. Pure Appl. Math. 36, 253–290, 571–593, 809–829.
R.I. McLachlan G.R.W. Quispel (2002) ArticleTitleSplitting methods Acta Numerica 11 341–434 Occurrence Handle2004f:34001 Occurrence Handle10.1017/S0962492902000053
R.I. McLachlan (1994) ArticleTitleSymplectic integration of Hamiltonian wave equations Numer. Math. 66 465–492 Occurrence Handle0831.65099 Occurrence Handle94m:65146
Preissmann, A. (1961). Propagation des intumescences dans les canaux et rivières. In First Congress French Assoc. for Computation, Grenoble.
S. Reich (2000) ArticleTitleFinite volume methods for multi-symplectic PDEs BIT 40 559–582 Occurrence Handle0965.65131 Occurrence Handle2002d:65130 Occurrence Handle10.1023/A:1022375915113
S. Reich (2000) ArticleTitleMulti-symplectic Runge–Kutta collocation methods for Hamiltonian wave equations J. Comput. Phys. 157 IssueID2 473–499 Occurrence Handle0946.65132 Occurrence Handle2001a:65087 Occurrence Handle10.1006/jcph.1999.6372
R.D. Richtmyer K.W. Morton (1967) Difference Methods for Initial-Value Problems Wiley New York
J.M. Sanz-Serna M.P. Calvo (1994) Numerical Hamiltonian Problems Chapman and Hall London
L.N. Trefethen (1997) ArticleTitlePseudospectra of linear operators SIAM Review 39 383–406 Occurrence Handle0896.15006 Occurrence Handle98i:47004 Occurrence Handle10.1137/S0036144595295284
C.V. Turner R.R. Rosales (1997) ArticleTitleThe small dispersion limit for a nonlinear semidiscrete system of equation Studies Appl. Math. 99 205–254 Occurrence Handle98d:35153
N.J. Zabusky M.D. Kruskal (1965) ArticleTitleInteraction of ‘solitons’ in a collisionless plasma and the recurrence of initial states Phys. Rev. 15 240–243
P.F. Zhao M.Z. Qin (2000) ArticleTitleMultisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation J. Phys. A 33 IssueID18 3613–3626 Occurrence Handle2001e:37094 Occurrence Handle10.1088/0305-4470/33/18/308
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Ascher, U.M., McLachlan, R.I. On Symplectic and Multisymplectic Schemes for the KdV Equation. J Sci Comput 25, 83–104 (2005). https://doi.org/10.1007/s10915-004-4634-6
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DOI: https://doi.org/10.1007/s10915-004-4634-6