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Multisymplectic geometry, variational integrators, and nonlinear PDEs. (English) Zbl 0951.70002
The article discusses the use of discrete variational principles for the construction of numerical integration methods for field theories. It extends Veselov’s approach to symplectic integrators to infinite-dimensional systems using multisymplectic geometry. The authors work throughout on the Lagrangian side. In fact, one important aspect of the article is to show how a (multi-)symplectic structure on the phase space can be obtained directly from variational principles. As a concrete example, a class of nonlinear wave equations containing the sine-Gordon equation, is analysed within this framework and integration methods for this class are derived. The quality of the new methods are assessed by numerically studying two soliton kink and antikink collisions. It turns out that the new methods compare favourably with other methods previously proposed in the literature.

MSC:
70-08Computational methods (mechanics of particles and systems)
70H30Other variational principles (mechanics of particles and systems)
70S05Lagrangian formalism and Hamiltonian formalism
49S05Variational principles of physics