zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Multi-symplectic methods for the Itô-type coupled KdV equation. (English) Zbl 1244.65232
Summary: We find that the Itô-type coupled Korteweg-de Vries (KdV) equation can be written as a multi-symplectic Hamiltonian partial differential equation. Then, multi-symplectic Fourier pseudospectral method and multi-symlpectic wavelet collocation method are constructed for this equation. In the numerical experiments, we show the effectiveness of the proposed methods. Some comparisons between the proposed methods are also made with respect to global conservation properties.

65P10Numerical methods for Hamiltonian systems including symplectic integrators
37M15Symplectic integrators (dynamical systems)
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q53KdV-like (Korteweg-de Vries) equations
65M70Spectral, collocation and related methods (IVP of PDE)
65T60Wavelets (numerical methods)
Full Text: DOI
[1] Ito, M.: Symmetries and conservation laws of a coupled nonlinear wave equation. Phys. lett. A 91, 405-420 (1982)
[2] Guha-Roy, C.: Solution of coupled KdV-type equations. Int. J. Theor. phys. 29, No. 8, 863-866 (1990) · Zbl 0711.35122
[3] Xu, Y.; Shu, C. -W.: Local discontinuous Galerkin methods for the Kuramoto -- Sivashinsky equations and the Itô-type coupled KdV equations. Comput. methods appl. Mech. eng. 195, 3430-3447 (2006) · Zbl 1124.76035
[4] Bridges, T. J.: Multi-symplectic structures and wave propagation. Math. proc. Camb. phil. Soc. 121, 147-190 (1997) · Zbl 0892.35123
[5] Bridges, T. J.; Reich, S.: Multi-symplectic integrators: numerical schemes for Hamiltonian pdes that conserve symplecticity. Phys. lett. A 284, 184-193 (2001) · Zbl 0984.37104
[6] Reich, S.: Multi-symplectic Runge -- Kutta collocation methods for Hamiltonian wave equations. J. comput. Phys. 157, 473-499 (2000) · Zbl 0946.65132
[7] Bridges, T. J.; Reich, S.: Numerical methods for Hamiltonian pdes. J. phys. A: math. Gen. 39, 5287-5320 (2006) · Zbl 1090.65138
[8] Bridges, T. J.; Reich, S.: Multi-symplectic spectral discretizations for the Zakharov -- Kuznetsov and shallow water equations. Physica D 152 -- 153, 491-504 (2001) · Zbl 1032.76053
[9] Chen, J.; Qin, M.: Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation. Electr. numer. Anal. 12, 193-204 (2001) · Zbl 0980.65108
[10] Moore, B.; Reich, S.: Backward error analysis for multi-symplectic integration methods. Numer. math. 95, 625-652 (2003) · Zbl 1033.65113
[11] Ryland, B. N.; Mclachlan, B. I.; Frank, J.: On multisymplecticity of partitioned Runge -- Kutta and splitting methods. Int. J. Comput. math. 84, 847-869 (2007) · Zbl 1125.65115
[12] Kong, L.; Hong, J.; Zang, J.: Splitting multi-symplectic integrators for Maxwell’s equation. J. comput. Phys. 229, 4259-4278 (2010) · Zbl 1192.78045
[13] Chen, Y.; Zhu, H.; Song, S.: Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation. Comput. phys. Commun. 181, 1231-1241 (2010) · Zbl 1219.65149
[14] Zhao, P.; Qin, M.: Multisymplectic geometry and multisymplectic preissman scheme for the KdV equation. J. phys. A: math. Gen. 33, 3613-3626 (2000) · Zbl 0989.37062
[15] Wang, Y.; Wang, B.; Qin, M.: Numerical implementation of the multisymplectic preissman scheme and its equivalent schemes. Appl. math. Comput. 149, 299-326 (2004) · Zbl 1047.65107
[16] Asher, U. M.; Mclachlan, R. I.: Multisymplectic box shemes and the Korteweg -- de Vries equation. Appl. numer. Math. 48, 255-269 (2004) · Zbl 1038.65138
[17] Asher, U. M.; Mclachlan, R. I.: On symplectic and multisymplectic shemes for the KdV equation. J. sci. Comput. 25, 83-104 (2005) · Zbl 1203.65277
[18] Aydın, A.; Karasözen, B.: Multisymplectic box schemes for the complex modified Korteweg -- de Vries equation. J. math. Phys. 51, 083511-083534 (2010) · Zbl 1312.35149
[19] Cai, J.: Multisymplectic numerical methods for the regularized long-wave equation. Comput. phys. Commun. 180, 1821-1831 (2009) · Zbl 1197.65144
[20] Zhu, H.; Tang, L.; Song, S.; Tang, Y.; Wang, D.: Symplectic wavelet collocation method for Hamiltonian wave equations. J. comput. Phys. 229, 2550-2572 (2010) · Zbl 1185.65194