31385
2006
2006
eng
article
1
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Numerical methods for Hamiltonian PDEs
The paper provides an introduction and survey of conservative discretization methods for Hamiltonian partial differential equations. The emphasis is on variational, symplectic and multi-symplectic methods. The derivation of methods as well as some of their fundamental geometric properties are discussed. Basic principles are illustrated by means of examples from wave and fluid dynamics
http://iopscience.iop.org/1751-8121/
10.1088/0305-4470/39/19/S02
0305-4470 - 39 (2006), 19, S
allegro:1991-2014
10101814
Journal of physics / A. - ISSN 0305-4470 - 39 (2006), 19, S. 5287 - 5320
Thomas J. Bridges
Sebastian Reich
Institut für Physik und Astronomie
Referiert
Institut für Physik
12018
2006
2006
eng
article
1
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Linear PDEs and numerical methods that preserve a multisymplectic conservation law
Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich [Phys. Lett. A, 284 ( 2001), pp. 184-193] and Reich [J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type restriction on Delta t/Delta x might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of Delta t/Delta x despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351-395]
http://epubs.siam.org/sisc/
10.1137/050628271
1064-8275
allegro:1991-2014
10102143
Siam journal on scientific computing. - ISSN 1064-8275. - 28 (2006), 1, S. 260 - 277
Jason Frank
Brian E. Moore
Sebastian Reich
Institut für Mathematik
Referiert