| Zentralblatt MATH - REVIEW SUBMISSION FORM |
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| Name: | ||||||||||||||||||
| Miloslav Znojil | ||||||||||||||||||
| Reviewer number: | ||||||||||||||||||
| 9689 | ||||||||||||||||||
| Email: | ||||||||||||||||||
| znojil@ujf.cas.cz | ||||||||||||||||||
| Item's zbl-Number: | ||||||||||||||||||
| DE 0164 5672 4 | ||||||||||||||||||
| Author(s): | ||||||||||||||||||
| Chen,Jing-Bo; Qin, Meng-Zhao | ||||||||||||||||||
| Shorttitle: | ||||||||||||||||||
| Multi-symplectic Fourier pseudospectral method for NLS | ||||||||||||||||||
| Source: | ||||||||||||||||||
| ETNA, Electron. Trans. Numer. Anal. 12, 193 - 204, elctronic only (2001) | ||||||||||||||||||
| Classification: | ||||||||||||||||||
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Keywords:
| Nonlinear Schroedinger equation; periodic boundary conditions; numericla solution; Fourier pseudospectral method; discrete version of multi-symplectic structure; two conservation laws; | Review: | Nonlinear Schroedinger equation on real space with periodic boundary conditions is interesting as a model with numerically induced chaos. Its numerical solution is considered using N equidistant collocation points in combination with Fourier transformation (also modified to fast Fourier transformation). As a key ingredient and mechanism of control of errors, the symplectic (multisymplectic) structure of the Hamiltonian system in question is employed, and it is required (and achieved) that N conservation laws exists after the discretization (in numerical experiments, the local energy conservation is being monitored for illustration and/or as a test of coarseness of the discretization). \par The algorithm proves efficient. Its idea is taken from an unpublished work by Bridges and Reich on similar equations (viz., Zakharov Kuznetsov equations and equations describing shallow-water in one dimension). Its implementation to the NLS problem is based on the presence of two pre-symplectic forms (characterized by the respective four-by-four skew-symmetric matrices M and K) and on the related energy conservation and momentum conservation laws. Technically, the resulting Runge Kutta discretization with conservation is rendered feasible by the successful evaluation of differentiation matrices at collocation points in closed form. Remarks to the editors: |
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