Dispersive properties of multisymplectic integrators
Journal of Computational Physics
Multi-symplectic integration of coupled non-linear Schrodinger system with soliton solutions
International Journal of Computer Mathematics
Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations
Mathematics and Computers in Simulation
Dispersion, group velocity, and multisymplectic discretizations
Mathematics and Computers in Simulation
Applied Numerical Mathematics
Symplectic and multisymplectic numerical methods for Maxwell's equations
Journal of Computational Physics
Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations
Journal of Computational Physics
Conformal conservation laws and geometric integration for damped Hamiltonian PDEs
Journal of Computational Physics
Journal of Computational Physics
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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].